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class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.12299">arXiv:2503.12299</a> <span> [<a href="https://arxiv.org/pdf/2503.12299">pdf</a>, <a href="https://arxiv.org/ps/2503.12299">ps</a>, <a href="https://arxiv.org/format/2503.12299">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Dual Murnaghan-Nakayama rule for Hecke algebras in Type $A$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yu Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.12299v1-abstract-short" style="display: inline;"> Let $蠂^位_渭$ be the value of the irreducible character $蠂^位$ of the Hecke algebra of the symmetric group on the conjugacy class of type $渭$. The usual Murnaghan-Nakayama rule provides an iterative algorithm based on reduction of the lower partition $渭$. In this paper, we establish a dual Murnaghan-Nakayama rule for Hecke algebras of type $A$ using vertex operators by applying reduction to the upper… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.12299v1-abstract-full').style.display = 'inline'; document.getElementById('2503.12299v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.12299v1-abstract-full" style="display: none;"> Let $蠂^位_渭$ be the value of the irreducible character $蠂^位$ of the Hecke algebra of the symmetric group on the conjugacy class of type $渭$. The usual Murnaghan-Nakayama rule provides an iterative algorithm based on reduction of the lower partition $渭$. In this paper, we establish a dual Murnaghan-Nakayama rule for Hecke algebras of type $A$ using vertex operators by applying reduction to the upper partition $位$. We formulate an explicit recursion of the dual Murnaghan-Nakayama rule by employing the combinatorial model of ``brick tabloids", which refines a previous result by two of us (J. Algebra 598 (2022), 24--47). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.12299v1-abstract-full').style.display = 'none'; document.getElementById('2503.12299v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8pp</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C08; Secondary: 17B69; 05E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.21211">arXiv:2502.21211</a> <span> [<a href="https://arxiv.org/pdf/2502.21211">pdf</a>, <a href="https://arxiv.org/ps/2502.21211">ps</a>, <a href="https://arxiv.org/format/2502.21211">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Parabolic presentations of Yangian in types $B$ and $C$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chang%2C+Z">Zhihua Chang</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+H">Haitao Ma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.21211v2-abstract-short" style="display: inline;"> We establish a parabolic presentation of the extended Yangian $\X(\mathfrak{g}_{N})$ associated with the Lie algebras $\mathfrak{g}_{N}$ of type $B$ and $C$, parameterized by a symmetric composition $谓$ of $N$. By formulating a block matrix version of the RTT presentation of $\X(\mathfrak{g}_{N})$, we systematically derive the generators and relations through the Gauss decomposition of the generat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.21211v2-abstract-full').style.display = 'inline'; document.getElementById('2502.21211v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.21211v2-abstract-full" style="display: none;"> We establish a parabolic presentation of the extended Yangian $\X(\mathfrak{g}_{N})$ associated with the Lie algebras $\mathfrak{g}_{N}$ of type $B$ and $C$, parameterized by a symmetric composition $谓$ of $N$. By formulating a block matrix version of the RTT presentation of $\X(\mathfrak{g}_{N})$, we systematically derive the generators and relations through the Gauss decomposition of the generator matrix in $谓$-block form. Furthermore, leveraging this parabolic presentation, we obtain a novel formula for the center of $\X(\mathfrak{g}_{N})$, offering new insights into its structure. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.21211v2-abstract-full').style.display = 'none'; document.getElementById('2502.21211v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">50 pp; updated references</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 17B37; secondary: 81R51 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.15586">arXiv:2502.15586</a> <span> [<a href="https://arxiv.org/pdf/2502.15586">pdf</a>, <a href="https://arxiv.org/ps/2502.15586">ps</a>, <a href="https://arxiv.org/format/2502.15586">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Skew odd orthogonal characters and interpolating Schur polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhijun Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Danxia Wang</a>, <a href="/search/math?searchtype=author&query=Ye%2C+C">Chang Ye</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.15586v1-abstract-short" style="display: inline;"> We introduce two vertex operators to realize skew odd orthogonal characters $so_{位/渭}(x^{\pm})$ and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method also gives new proofs of the Jacobi--Trudi identity and Gelfand--Tsetlin patterns for $so_{位/渭}(x^{\pm})$. Moreover, combining the vertex operators related to characters… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.15586v1-abstract-full').style.display = 'inline'; document.getElementById('2502.15586v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.15586v1-abstract-full" style="display: none;"> We introduce two vertex operators to realize skew odd orthogonal characters $so_{位/渭}(x^{\pm})$ and derive the Cauchy identity for the skew characters via Toeplitz-Hankel-type determinant similar to the Schur functions. The method also gives new proofs of the Jacobi--Trudi identity and Gelfand--Tsetlin patterns for $so_{位/渭}(x^{\pm})$. Moreover, combining the vertex operators related to characters of types $C,D$ (\cite{Ba1996,JN2015}) and the new vertex operators related to $B$-type characters, we obtain three families of symmetric polynomials that interpolate among characters of $SO_{2n+1}(\mathbb{C})$, $SO_{2n}(\mathbb{C})$ and $Sp_{2n}(\mathbb{C})$, Their transition formulas are also explicitly given among symplectic and/or orthogonal characters and odd orthogonal characters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.15586v1-abstract-full').style.display = 'none'; document.getElementById('2502.15586v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Appendix with Xinyu Pan; 18pp</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E05; Secondary: 17B37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.19385">arXiv:2412.19385</a> <span> [<a href="https://arxiv.org/pdf/2412.19385">pdf</a>, <a href="https://arxiv.org/ps/2412.19385">ps</a>, <a href="https://arxiv.org/format/2412.19385">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+L">Li Zheng</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.19385v2-abstract-short" style="display: inline;"> We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\gl}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.19385v2-abstract-full').style.display = 'inline'; document.getElementById('2412.19385v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.19385v2-abstract-full" style="display: none;"> We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\gl}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.19385v2-abstract-full').style.display = 'none'; document.getElementById('2412.19385v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17pp</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 17B37; Secondary: 81R50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.18793">arXiv:2412.18793</a> <span> [<a href="https://arxiv.org/pdf/2412.18793">pdf</a>, <a href="https://arxiv.org/ps/2412.18793">ps</a>, <a href="https://arxiv.org/format/2412.18793">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> On character values of $GL_n(\mathbb F_q)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yu Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.18793v1-abstract-short" style="display: inline;"> In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $蠁$ of linear characters of $\overline{\mathbb F}_q$ under the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.18793v1-abstract-full').style.display = 'inline'; document.getElementById('2412.18793v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.18793v1-abstract-full" style="display: none;"> In this paper, we use vertex operator techniques to compute character values on unipotent classes of $\GL_n(\mathbb F_q)$. By realizing the Grothendieck ring $R_G=\bigoplus_{n\geq0}^\infty R(\GL_n(\mathbb F_q))$ as Fock spaces, we formulate the Murnanghan-Nakayama rule of $\GL_n(\mathbb F_q)$ between Schur functions colored by an orbit $蠁$ of linear characters of $\overline{\mathbb F}_q$ under the Frobenius automorphism on and modified Hall-Littlewood functions colored by $f_1=t-1$, which provides detailed information on the character table of $\GL_n(\mathbb F_q)$. As applications, we use vertex algebraic methods to determine the Steinberg characters of $\GL_n(\mathbb F_q)$, which were previously determined by Curtis-Lehrer-Tits via geometry of homology groups of spherical buildings and Springer-Zelevinsky utilizing Hopf algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.18793v1-abstract-full').style.display = 'none'; document.getElementById('2412.18793v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15pp</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C33; 17B69; Secondary: 05E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.03159">arXiv:2409.03159</a> <span> [<a href="https://arxiv.org/pdf/2409.03159">pdf</a>, <a href="https://arxiv.org/ps/2409.03159">ps</a>, <a href="https://arxiv.org/format/2409.03159">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/00927872.2024.2343750">10.1080/00927872.2024.2343750 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Whittaker modules for a subalgebra of N=2 superconformal algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Xu%2C+P">Pengfa Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.03159v1-abstract-short" style="display: inline;"> In this paper, Whittaker modules are studied for a subalgebra $\mathfrak{q}_蔚$ of the $\emph{N}$=2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.03159v1-abstract-full" style="display: none;"> In this paper, Whittaker modules are studied for a subalgebra $\mathfrak{q}_蔚$ of the $\emph{N}$=2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.03159v1-abstract-full').style.display = 'none'; document.getElementById('2409.03159v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Comm. Algebra 52 (2024), no. 10, 4161-4179 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01479">arXiv:2409.01479</a> <span> [<a href="https://arxiv.org/pdf/2409.01479">pdf</a>, <a href="https://arxiv.org/ps/2409.01479">ps</a>, <a href="https://arxiv.org/format/2409.01479">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Plethysm Stability of Schur's $Q$-functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Graf%2C+J">John Graf</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.01479v1-abstract-short" style="display: inline;"> Schur functions have been shown to satisfy certain stability properties and recurrence relations. In this paper, we prove analogs of these properties with Schur's $Q$-functions using vertex operator methods. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01479v1-abstract-full" style="display: none;"> Schur functions have been shown to satisfy certain stability properties and recurrence relations. In this paper, we prove analogs of these properties with Schur's $Q$-functions using vertex operator methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01479v1-abstract-full').style.display = 'none'; document.getElementById('2409.01479v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05; 05E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.09855">arXiv:2408.09855</a> <span> [<a href="https://arxiv.org/pdf/2408.09855">pdf</a>, <a href="https://arxiv.org/ps/2408.09855">ps</a>, <a href="https://arxiv.org/format/2408.09855">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> The $q$-immanants and higher quantum Capelli identities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a>, <a href="/search/math?searchtype=author&query=Molev%2C+A">Alexander Molev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.09855v2-abstract-short" style="display: inline;"> We construct polynomials ${\mathbb{S}}_渭(z)$ parameterized by Young diagrams $渭$, whose coefficients are central elements of the quantized enveloping algebra ${\rm U}_q({\mathfrak{gl}}_n)$. Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of $z$, we get $q$-analogues of Okounkov's quantum immanants f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09855v2-abstract-full').style.display = 'inline'; document.getElementById('2408.09855v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.09855v2-abstract-full" style="display: none;"> We construct polynomials ${\mathbb{S}}_渭(z)$ parameterized by Young diagrams $渭$, whose coefficients are central elements of the quantized enveloping algebra ${\rm U}_q({\mathfrak{gl}}_n)$. Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of $z$, we get $q$-analogues of Okounkov's quantum immanants for ${\mathfrak{gl}}_n$. We show that the Harish-Chandra image of ${\mathbb{S}}_渭(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09855v2-abstract-full').style.display = 'none'; document.getElementById('2408.09855v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, more detailed proofs are given, references extended</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.04921">arXiv:2408.04921</a> <span> [<a href="https://arxiv.org/pdf/2408.04921">pdf</a>, <a href="https://arxiv.org/ps/2408.04921">ps</a>, <a href="https://arxiv.org/format/2408.04921">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Irreducible characters of the generalized symmetric group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+H">Huimin Gao</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.04921v2-abstract-short" style="display: inline;"> The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.04921v2-abstract-full').style.display = 'inline'; document.getElementById('2408.04921v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.04921v2-abstract-full" style="display: none;"> The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of $C_k\wr S_n$ and modular characters of $S_{kn}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.04921v2-abstract-full').style.display = 'none'; document.getElementById('2408.04921v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages; code appended. Updated references</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C08; Secondary: 05E10; 17B69 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.04340">arXiv:2408.04340</a> <span> [<a href="https://arxiv.org/pdf/2408.04340">pdf</a>, <a href="https://arxiv.org/ps/2408.04340">ps</a>, <a href="https://arxiv.org/format/2408.04340">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/jlms.70114">10.1112/jlms.70114 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Twisted q-Yangians and Sklyanin determinants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.04340v3-abstract-short" style="display: inline;"> $q$-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the $q$-Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.04340v3-abstract-full').style.display = 'inline'; document.getElementById('2408.04340v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.04340v3-abstract-full" style="display: none;"> $q$-Yangians can be viewed both as quantum deformations of the loop algebras of upper triangular Lie algebras and deformations of the Yangian algebras. In this paper, we study the quantum affine algebra as a product of two copies of the $q$-Yangian algebras. This viewpoint enables us to investigate the invariant theory of quantum affine algebras and their twisted versions. We introduce the twisted Sklyanin determinant for twisted quantum affine algebras and establish various identities for the Sklyanin determinants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.04340v3-abstract-full').style.display = 'none'; document.getElementById('2408.04340v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages; updated version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 17B37; Secondary: 17B10; 81R50; 14M17; 15A15; 13A50 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. London Math. Soc. (2) 111 (2025) no.3, e70114 (40pp) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.00406">arXiv:2407.00406</a> <span> [<a href="https://arxiv.org/pdf/2407.00406">pdf</a>, <a href="https://arxiv.org/ps/2407.00406">ps</a>, <a href="https://arxiv.org/format/2407.00406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/5.0229568">10.1063/5.0229568 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> $RLL$-Realization and Its Hopf Superalgebra Structure of $U_{p, q}(\widehat{\mathfrak{gl}(m|n))}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+N">Naihong Hu</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Zhong%2C+X">Xin Zhong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.00406v3-abstract-short" style="display: inline;"> In this paper, we extend the Reshetikhin-Semenov-Tian-Shansky formulation of quantum affine algebras to the two-parameter quantum affine superalgebra $U_{p, q}(\widehat{\mathfrak{gl}}(m|n))$ and obtain its Drinfeld realization. We also derive its Hopf algebra structure by providing Drinfeld-type coproduct for the Drinfeld generators. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.00406v3-abstract-full" style="display: none;"> In this paper, we extend the Reshetikhin-Semenov-Tian-Shansky formulation of quantum affine algebras to the two-parameter quantum affine superalgebra $U_{p, q}(\widehat{\mathfrak{gl}}(m|n))$ and obtain its Drinfeld realization. We also derive its Hopf algebra structure by providing Drinfeld-type coproduct for the Drinfeld generators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.00406v3-abstract-full').style.display = 'none'; document.getElementById('2407.00406v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, update</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 17B37; Secondary 16T05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Phys. 65, 123501 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.00581">arXiv:2406.00581</a> <span> [<a href="https://arxiv.org/pdf/2406.00581">pdf</a>, <a href="https://arxiv.org/ps/2406.00581">ps</a>, <a href="https://arxiv.org/format/2406.00581">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On a Pieri-like rule for the Petrie symmetric functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jin%2C+E+Y">Emma Yu Jin</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.00581v2-abstract-short" style="display: inline;"> A $k$-ribbon tiling is a decomposition of a connected skew diagram into disjoint ribbons of size $k$. In this paper, we establish a connection between a subset of $k$-ribbon tilings and Petrie symmetric functions, thus providing a combinatorial interpretation for the coefficients in a Pieri-like rule for the Petrie symmetric functions due to Grinberg (Algebr. Comb. 5 (2022), no. 5, 947-1013). This… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00581v2-abstract-full').style.display = 'inline'; document.getElementById('2406.00581v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.00581v2-abstract-full" style="display: none;"> A $k$-ribbon tiling is a decomposition of a connected skew diagram into disjoint ribbons of size $k$. In this paper, we establish a connection between a subset of $k$-ribbon tilings and Petrie symmetric functions, thus providing a combinatorial interpretation for the coefficients in a Pieri-like rule for the Petrie symmetric functions due to Grinberg (Algebr. Comb. 5 (2022), no. 5, 947-1013). This also extends a result by Cheng, Chou and Eu et al. (Proc. Amer. Math. Soc. 151 (2023), no. 5, 1839-1854). As a bonus, our findings can be effectively utilized to derive certain specializations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00581v2-abstract-full').style.display = 'none'; document.getElementById('2406.00581v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Added some figures and details. 17 pages, 9 figures. All comments are welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05; 05A17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.13137">arXiv:2405.13137</a> <span> [<a href="https://arxiv.org/pdf/2405.13137">pdf</a>, <a href="https://arxiv.org/ps/2405.13137">ps</a>, <a href="https://arxiv.org/format/2405.13137">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2025.02.002">10.1016/j.jalgebra.2025.02.002 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Pfaffian Formulation of Schur's $Q$-functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Graf%2C+J">John Graf</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.13137v1-abstract-short" style="display: inline;"> We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_位$ to be indexed by compositions $位$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing $Q_位$ into sums indexed by partitions with remove… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13137v1-abstract-full').style.display = 'inline'; document.getElementById('2405.13137v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.13137v1-abstract-full" style="display: none;"> We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_位$ to be indexed by compositions $位$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing $Q_位$ into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's $Q$-functions using only simple algebraic methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13137v1-abstract-full').style.display = 'none'; document.getElementById('2405.13137v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 669 (2025) 1--25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.06597">arXiv:2405.06597</a> <span> [<a href="https://arxiv.org/pdf/2405.06597">pdf</a>, <a href="https://arxiv.org/ps/2405.06597">ps</a>, <a href="https://arxiv.org/format/2405.06597">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219498826501173">10.1142/S0219498826501173 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> $RLL$-realization of two-parameter quantum affine algebra in type $C_n^{(1)}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhong%2C+X">Xin Zhong</a>, <a href="/search/math?searchtype=author&query=Hu%2C+N">Naihong Hu</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.06597v2-abstract-short" style="display: inline;"> We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization 谩 la Faddeev, Reshetikhin and Takhtajan. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.06597v2-abstract-full" style="display: none;"> We establish an explicit correspondence between the Drinfeld current algebra presentation for the two-parameter quantum affine algebra $U_{r, s}(\mathrm{C}_n^{(1)})$ and the $R$-matrix realization 谩 la Faddeev, Reshetikhin and Takhtajan. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.06597v2-abstract-full').style.display = 'none'; document.getElementById('2405.06597v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra and its Applications (2026) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.08599">arXiv:2402.08599</a> <span> [<a href="https://arxiv.org/pdf/2402.08599">pdf</a>, <a href="https://arxiv.org/ps/2402.08599">ps</a>, <a href="https://arxiv.org/format/2402.08599">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> On an optimal problem of bilinear forms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yibo Liu</a>, <a href="/search/math?searchtype=author&query=Sun%2C+J">Jiacheng Sun</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+C">Chengrui Zhao</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+H">Haoran Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.08599v1-abstract-short" style="display: inline;"> We study an optimization problem originated from the Grothendieck constant. A generalized normal equation is proposed and analyzed. We establish a correspondence between solutions of the general normal equation and its dual equation. Explicit solutions are described for the two-dimensional case. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.08599v1-abstract-full" style="display: none;"> We study an optimization problem originated from the Grothendieck constant. A generalized normal equation is proposed and analyzed. We establish a correspondence between solutions of the general normal equation and its dual equation. Explicit solutions are described for the two-dimensional case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.08599v1-abstract-full').style.display = 'none'; document.getElementById('2402.08599v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6pp</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13815">arXiv:2312.13815</a> <span> [<a href="https://arxiv.org/pdf/2312.13815">pdf</a>, <a href="https://arxiv.org/ps/2312.13815">ps</a>, <a href="https://arxiv.org/format/2312.13815">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/5.0211081">10.1063/5.0211081 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Eigenvalues of quantum Gelfand invariants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a>, <a href="/search/math?searchtype=author&query=Molev%2C+A">Alexander Molev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13815v4-abstract-short" style="display: inline;"> We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev (1989). We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for ${\mathfrak {gl}}_n$. The calculation is based on Liouville-type formulas relating two families of central elements in the quan… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13815v4-abstract-full').style.display = 'inline'; document.getElementById('2312.13815v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13815v4-abstract-full" style="display: none;"> We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin, Takhtadzhyan and Faddeev (1989). We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for ${\mathfrak {gl}}_n$. The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type $A$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13815v4-abstract-full').style.display = 'none'; document.getElementById('2312.13815v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, construction of central elements and some proofs were simplified in v2, limit values as q->1 are discussed in v3, reference to earlier work on Liouville formula added in v4</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Math. Phys. 65, 061703 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13681">arXiv:2312.13681</a> <span> [<a href="https://arxiv.org/pdf/2312.13681">pdf</a>, <a href="https://arxiv.org/ps/2312.13681">ps</a>, <a href="https://arxiv.org/format/2312.13681">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Irreducible characters and bitrace for the $q$-rook monoid </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yu Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13681v1-abstract-short" style="display: inline;"> This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13681v1-abstract-full').style.display = 'inline'; document.getElementById('2312.13681v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13681v1-abstract-full" style="display: none;"> This paper studies irreducible characters of the $q$-rook monoid algebra $R_n(q)$ using the vertex algebraic method. Based on the Frobenius formula for $R_n(q)$, a new iterative character formula is derived with the help of the vertex operator realization of the Schur symmetric function. The same idea also leads to a simple proof of the Murnaghan-Nakayama rule for $R_n(q)$. We also introduce the bitrace for the $q$-monoid and obtain a general combinatorial formula for the bitrace, which generalizes the counterpart for the Iwahori-Hecke algebra. The character table of $R_n(q)$ with $|渭|=5$ is listed in the appendix. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13681v1-abstract-full').style.display = 'none'; document.getElementById('2312.13681v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C08; Secondary: 17B69; 05E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.13675">arXiv:2312.13675</a> <span> [<a href="https://arxiv.org/pdf/2312.13675">pdf</a>, <a href="https://arxiv.org/ps/2312.13675">ps</a>, <a href="https://arxiv.org/format/2312.13675">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00026-023-00686-8">10.1007/s00026-023-00686-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A spin analog of the plethystic Murnaghan-Nakayama rule </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cao%2C+Y">Yue Cao</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.13675v1-abstract-short" style="display: inline;"> As a spin analog of the plethystic Murnaghan-Nakayama rule for Schur functions, the plethystic Murnaghan-Nakayama rule for Schur $Q$-functions is established with the help of the vertex operator realization. This generalizes both the Murnaghan-Nakayama rule and the Pieri rule for Schur $Q$-functions. A plethystic Murnaghan-Nakayama rule for Hall-Littlewood functions is also investigated. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.13675v1-abstract-full" style="display: none;"> As a spin analog of the plethystic Murnaghan-Nakayama rule for Schur functions, the plethystic Murnaghan-Nakayama rule for Schur $Q$-functions is established with the help of the vertex operator realization. This generalizes both the Murnaghan-Nakayama rule and the Pieri rule for Schur $Q$-functions. A plethystic Murnaghan-Nakayama rule for Hall-Littlewood functions is also investigated. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.13675v1-abstract-full').style.display = 'none'; document.getElementById('2312.13675v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ann. Combin. 28 (2024) 655--679 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.15730">arXiv:2310.15730</a> <span> [<a href="https://arxiv.org/pdf/2310.15730">pdf</a>, <a href="https://arxiv.org/ps/2310.15730">ps</a>, <a href="https://arxiv.org/format/2310.15730">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jcta2024.105920">10.1016/j.jcta2024.105920 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2310.15730v1-abstract-short" style="display: inline;"> We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters $q,t$ (denoted by $螞[q,t]$) are computed by assigning some values to skew Macdonald polynomials in $位$-ring notation. The new rule is utilized to provide new ite… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.15730v1-abstract-full').style.display = 'inline'; document.getElementById('2310.15730v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.15730v1-abstract-full" style="display: none;"> We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters $q,t$ (denoted by $螞[q,t]$) are computed by assigning some values to skew Macdonald polynomials in $位$-ring notation. The new rule is utilized to provide new iterative formulas and also recover various existing formulas in a unified manner. Specifically the following applications are discussed: (i) A $(q,t)$-Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the $q$-Murnaghan-Nakayama rule; (ii) An iterative formula for the $(q,t)$-Green polynomial is deduced; (iii) A simple proof of the Murnaghan-Nakayama rule for the Hecke algebra and the Hecke-Clifford algebra is offered; (iv) A combinatorial inversion of the Pieri rule for Hall-Littlewood functions is derived with the help of the vertex operator realization of the Hall-Littlewood functions; (v) Two iterative formulae for the $(q,t)$-Kostka polynomials $K_{位渭}(q,t)$ are obtained from the dual version of our multiparametric Murnaghan-Nakayama rule, one of which yields an explicit formula for arbitrary $位$ and $渭$ in terms of the generalized $(q, t)$-binomial coefficient introduced independently by Lassalle and Okounkov. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.15730v1-abstract-full').style.display = 'none'; document.getElementById('2310.15730v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 pp, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E05; 05E10; Secondary: 17B69; 20C08; 15A66 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Comb. Theory A 207 (2024), 10592032 (34pp) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.15330">arXiv:2309.15330</a> <span> [<a href="https://arxiv.org/pdf/2309.15330">pdf</a>, <a href="https://arxiv.org/ps/2309.15330">ps</a>, <a href="https://arxiv.org/format/2309.15330">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2024.04.020">10.1016/j.jalgebra.2024.04.020 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Characters of $GL_n(\mathbb F_q)$ and vertex operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yu Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.15330v4-abstract-short" style="display: inline;"> In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group $\GL_n(\mathbb F_q)$. Green's theory of $\GL_n(\mathbb F_q)$ is recovered and enhanced under the realization of the Grothendieck ring of representations $R_G=\bigoplus_{n\geq 0}R(\GL_n(\mathbb F_q))$ as two isomorphic Fock spaces associated to two infinite-di… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.15330v4-abstract-full').style.display = 'inline'; document.getElementById('2309.15330v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.15330v4-abstract-full" style="display: none;"> In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group $\GL_n(\mathbb F_q)$. Green's theory of $\GL_n(\mathbb F_q)$ is recovered and enhanced under the realization of the Grothendieck ring of representations $R_G=\bigoplus_{n\geq 0}R(\GL_n(\mathbb F_q))$ as two isomorphic Fock spaces associated to two infinite-dimensional $F$-equivariant Heisenberg Lie algebras $\widehat{\mathfrak{h}}_{\hat{\overline{\mathbb F}}_q}$ and $\widehat{\mathfrak{h}}_{\overline{\mathbb F}_q}$, where $F$ is the Frobenius automorphism of the algebraically closed field $\overline{\mathbb F}_q$. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space $R_G$ as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.15330v4-abstract-full').style.display = 'none'; document.getElementById('2309.15330v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, one chart; Final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C33; 17B69; Secondary: 05E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 653 (2024), 109-132 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.00980">arXiv:2309.00980</a> <span> [<a href="https://arxiv.org/pdf/2309.00980">pdf</a>, <a href="https://arxiv.org/ps/2309.00980">ps</a>, <a href="https://arxiv.org/format/2309.00980">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219498825501713">10.1142/S0219498825501713 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Kostant's generating functions and McKay-Slodowy correspondence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhijun Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Danxia Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.00980v3-abstract-short" style="display: inline;"> Let $N\unlhd G$ be a pair of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant's generating functions for the decomposition of the $\mathrm{SL}_2(\mathbb C)$-module $S^k(V)$ restricted to $N\lhd G$ in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.00980v3-abstract-full').style.display = 'inline'; document.getElementById('2309.00980v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.00980v3-abstract-full" style="display: none;"> Let $N\unlhd G$ be a pair of finite subgroups of $\mathrm{SL}_2(\mathbb{C})$ and $V$ a finite-dimensional fundamental $G$-module. We study Kostant's generating functions for the decomposition of the $\mathrm{SL}_2(\mathbb C)$-module $S^k(V)$ restricted to $N\lhd G$ in connection with the McKay-Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincar茅 series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.00980v3-abstract-full').style.display = 'none'; document.getElementById('2309.00980v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra Appl. (2025) No. 7, 2550171 (15pp) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.14573">arXiv:2307.14573</a> <span> [<a href="https://arxiv.org/pdf/2307.14573">pdf</a>, <a href="https://arxiv.org/ps/2307.14573">ps</a>, <a href="https://arxiv.org/format/2307.14573">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> General Capelli-type identities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yinlong Liu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.14573v2-abstract-short" style="display: inline;"> The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.14573v2-abstract-full').style.display = 'inline'; document.getElementById('2307.14573v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.14573v2-abstract-full" style="display: none;"> The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of $U(gl(n))$ which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for $U(gl(n))$ in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.14573v2-abstract-full').style.display = 'none'; document.getElementById('2307.14573v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 17B37 Secondary: 20G05; 17B35; 17B66; 05E10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.07439">arXiv:2304.07439</a> <span> [<a href="https://arxiv.org/pdf/2304.07439">pdf</a>, <a href="https://arxiv.org/ps/2304.07439">ps</a>, <a href="https://arxiv.org/format/2304.07439">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00605-023-01843-0">10.1007/s00605-023-01843-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Q-Kostka polynomials and spin Green polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+A">Anguo Jiang</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2304.07439v1-abstract-short" style="display: inline;"> We study the $Q$-Kostka polynomials $L_{位渭}(t)$ by the vertex operator realization of the $Q$-Hall-Littlewood functions $G_位(x;t)$ and derive new formulae for $L_{位渭}(t)$. In particular, we have established stability property for the Q-Kostka polynomials. We also introduce spin Green polynomials $Y^位_渭(t)$ as both an analogue of the Green polynomials and deformation of the spin irreducible charact… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.07439v1-abstract-full').style.display = 'inline'; document.getElementById('2304.07439v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.07439v1-abstract-full" style="display: none;"> We study the $Q$-Kostka polynomials $L_{位渭}(t)$ by the vertex operator realization of the $Q$-Hall-Littlewood functions $G_位(x;t)$ and derive new formulae for $L_{位渭}(t)$. In particular, we have established stability property for the Q-Kostka polynomials. We also introduce spin Green polynomials $Y^位_渭(t)$ as both an analogue of the Green polynomials and deformation of the spin irreducible characters of $\mathfrak S_n$. Iterative formulas of the spin Green polynomials are given and some favorable properties parallel to the Green polynomials are obtained. Tables of $Y^位_渭(t)$ are included for $n\leq7.$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.07439v1-abstract-full').style.display = 'none'; document.getElementById('2304.07439v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E05; Secondary: 17B69; 05E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Monatsh. Math. 201 (2023), no. 1, 109-125 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.14459">arXiv:2303.14459</a> <span> [<a href="https://arxiv.org/pdf/2303.14459">pdf</a>, <a href="https://arxiv.org/ps/2303.14459">ps</a>, <a href="https://arxiv.org/format/2303.14459">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1093/imrn/rnad158">10.1093/imrn/rnad158 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Murnaghan-Nakayama rule and spin bitrace for the Hecke-Clifford Algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.14459v2-abstract-short" style="display: inline;"> A Pfaffian-type Murnaghan-Nakayama rule is derived for the Hecke-Clifford algebra $\mathcal{H}^c_n$ based on the Frobenius formula and vertex operators, and this leads to a combinatorial version via the tableaux realization of Schur's $Q$-functions. As a consequence, a general formula for the irreducible characters $味^{\la}_渭(q)$ using partition-valued functions is derived. Meanwhile, an iterative… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.14459v2-abstract-full').style.display = 'inline'; document.getElementById('2303.14459v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.14459v2-abstract-full" style="display: none;"> A Pfaffian-type Murnaghan-Nakayama rule is derived for the Hecke-Clifford algebra $\mathcal{H}^c_n$ based on the Frobenius formula and vertex operators, and this leads to a combinatorial version via the tableaux realization of Schur's $Q$-functions. As a consequence, a general formula for the irreducible characters $味^{\la}_渭(q)$ using partition-valued functions is derived. Meanwhile, an iterative formula on the indexing partition $\la$ via the Pieri rule is also deduced. As applications, some compact formulae of the irreducible characters are given for special partitions and a symmetric property of the irreducible character is found. We also introduce the spin bitrace as the analogue of the bitrace for the Hecke algebra and derive its general combinatorial formula. Tables of irreducible characters are listed for $n\leq7.$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.14459v2-abstract-full').style.display = 'none'; document.getElementById('2303.14459v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages, 5 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 20C08; 15A66; Secondary: 17B69; 20C15; 05E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. Mat. Res. Not. IMRN 19 (2023), 17060-17099 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.12608">arXiv:2303.12608</a> <span> [<a href="https://arxiv.org/pdf/2303.12608">pdf</a>, <a href="https://arxiv.org/ps/2303.12608">ps</a>, <a href="https://arxiv.org/format/2303.12608">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2023.06.002">10.1016/j.jalgebra.2023.06.002 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum algebra of multiparameter Manin matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yinlong Liu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.12608v2-abstract-short" style="display: inline;"> Multiparametric quantum semigroups $\mathrm{M}_{\hat{q}, \hat{p}}(n)$ are generalization of the one-parameter general linear semigroups $\mathrm{M}_q(n)$, where $\hat{q}=(q_{ij})$ and $\hat{p}=(p_{ij})$ are $2n^2$ parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain se… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.12608v2-abstract-full').style.display = 'inline'; document.getElementById('2303.12608v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.12608v2-abstract-full" style="display: none;"> Multiparametric quantum semigroups $\mathrm{M}_{\hat{q}, \hat{p}}(n)$ are generalization of the one-parameter general linear semigroups $\mathrm{M}_q(n)$, where $\hat{q}=(q_{ij})$ and $\hat{p}=(p_{ij})$ are $2n^2$ parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also generalized to multiparameter $q$-Yangians. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.12608v2-abstract-full').style.display = 'none'; document.getElementById('2303.12608v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages; final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E10; Secondary: 17B37; 58A17; 15A75; 15B33; 15A15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 655 (2024), 586-618 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.10664">arXiv:2303.10664</a> <span> [<a href="https://arxiv.org/pdf/2303.10664">pdf</a>, <a href="https://arxiv.org/ps/2303.10664">ps</a>, <a href="https://arxiv.org/format/2303.10664">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/pjm.2023.325.127">10.2140/pjm.2023.325.127 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Spin Kostka polynomials and vertex operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.10664v2-abstract-short" style="display: inline;"> An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{尉渭}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an applicat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.10664v2-abstract-full').style.display = 'inline'; document.getElementById('2303.10664v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.10664v2-abstract-full" style="display: none;"> An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{尉渭}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{尉渭}(t)$ for $|尉|\leq6$ are listed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.10664v2-abstract-full').style.display = 'none'; document.getElementById('2303.10664v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 5 tables (correction of authors' names)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E10; Secondary: 17B69; 05E05 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Pacific J. Math. 325 (2023) 127-146 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.04732">arXiv:2301.04732</a> <span> [<a href="https://arxiv.org/pdf/2301.04732">pdf</a>, <a href="https://arxiv.org/ps/2301.04732">ps</a>, <a href="https://arxiv.org/format/2301.04732">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2023.10.002">10.1016/j.jalgebra.2023.10.002 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Semi-infinite construction for the double Yangian of type $A_1^{(1)}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Butorac%2C+M">Marijana Butorac</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Ko%C5%BEi%C4%87%2C+S">Slaven Ko啪i膰</a>, <a href="/search/math?searchtype=author&query=Yang%2C+F">Fan Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.04732v1-abstract-short" style="display: inline;"> We consider certain infinite dimensional modules of level 1 for the double Yangian $\text{DY}(\mathfrak{gl}_2)$ which are based on the Iohara-Kohno realization. We show that they possess topological bases of Feigin-Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.04732v1-abstract-full').style.display = 'inline'; document.getElementById('2301.04732v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.04732v1-abstract-full" style="display: none;"> We consider certain infinite dimensional modules of level 1 for the double Yangian $\text{DY}(\mathfrak{gl}_2)$ which are based on the Iohara-Kohno realization. We show that they possess topological bases of Feigin-Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give some applications of these bases to the representation theory of the corresponding quantum affine vertex algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.04732v1-abstract-full').style.display = 'none'; document.getElementById('2301.04732v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 638 (2024), 465-487 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.11435">arXiv:2212.11435</a> <span> [<a href="https://arxiv.org/pdf/2212.11435">pdf</a>, <a href="https://arxiv.org/ps/2212.11435">ps</a>, <a href="https://arxiv.org/format/2212.11435">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.aim.2024.109907">10.1016/j.aim.2024.109907 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum Sugawara operators in type $A$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a>, <a href="/search/math?searchtype=author&query=Molev%2C+A">Alexander Molev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.11435v1-abstract-short" style="display: inline;"> We construct Sugawara operators for the quantum affine algebra of type $A$ in an explicit form. The operators are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This generalizes a previous construction (2016) where one-column diagrams were considered. We calculate the Harish-Chandra images of the Sugawara operators and identify them with the eigenva… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.11435v1-abstract-full').style.display = 'inline'; document.getElementById('2212.11435v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.11435v1-abstract-full" style="display: none;"> We construct Sugawara operators for the quantum affine algebra of type $A$ in an explicit form. The operators are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This generalizes a previous construction (2016) where one-column diagrams were considered. We calculate the Harish-Chandra images of the Sugawara operators and identify them with the eigenvalues of the operators acting in the $q$-deformed Wakimoto modules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.11435v1-abstract-full').style.display = 'none'; document.getElementById('2212.11435v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, extending the construction of arXiv:1505.03667 to arbitrary Young diagrams</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv. Math. 454 (2024), 109907 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.08412">arXiv:2212.08412</a> <span> [<a href="https://arxiv.org/pdf/2212.08412">pdf</a>, <a href="https://arxiv.org/ps/2212.08412">ps</a>, <a href="https://arxiv.org/format/2212.08412">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Plethystic Murnaghan-Nakayama rule via vertex operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cao%2C+Y">Yue Cao</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.08412v2-abstract-short" style="display: inline;"> Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan--Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of $s_谓$ in the plethysm product $(p_{n}\circ h_{k})s_渭$. Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.08412v2-abstract-full').style.display = 'inline'; document.getElementById('2212.08412v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.08412v2-abstract-full" style="display: none;"> Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan--Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of $s_谓$ in the plethysm product $(p_{n}\circ h_{k})s_渭$. Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.08412v2-abstract-full').style.display = 'none'; document.getElementById('2212.08412v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 5 figures. revised version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E10; 05E05; Secondary: 17B69 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2212.01895">arXiv:2212.01895</a> <span> [<a href="https://arxiv.org/pdf/2212.01895">pdf</a>, <a href="https://arxiv.org/ps/2212.01895">ps</a>, <a href="https://arxiv.org/format/2212.01895">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Twisted quantum affine algebras and equivariant $蠁$-coordinated modules for quantum vertex algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Kong%2C+F">Fei Kong</a>, <a href="/search/math?searchtype=author&query=Li%2C+H">Haisheng Li</a>, <a href="/search/math?searchtype=author&query=Tan%2C+S">Shaobin Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2212.01895v1-abstract-short" style="display: inline;"> This paper is about establishing a natural connection of quantum affine algebras with quantum vertex algebras. Among the main results, we establish $\hbar$-adic versions of the smash product construction of quantum vertex algebras and their $蠁$-coordinated quasi modules, which were obtained before in a sequel, we construct a family of $\hbar$-adic quantum vertex algebras $V_L[[\hbar]]^畏$ as defo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.01895v1-abstract-full').style.display = 'inline'; document.getElementById('2212.01895v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2212.01895v1-abstract-full" style="display: none;"> This paper is about establishing a natural connection of quantum affine algebras with quantum vertex algebras. Among the main results, we establish $\hbar$-adic versions of the smash product construction of quantum vertex algebras and their $蠁$-coordinated quasi modules, which were obtained before in a sequel, we construct a family of $\hbar$-adic quantum vertex algebras $V_L[[\hbar]]^畏$ as deformations of the lattice vertex algebras $V_L$, and establish a natural connection between twisted quantum affine algebras of type $A, D, E$ and equivariant $蠁$-coordinated quasi modules for the $\hbar$-adic quantum vertex algebras $V_L[[\hbar]]^畏$ with certain specialized $畏$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2212.01895v1-abstract-full').style.display = 'none'; document.getElementById('2212.01895v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.00288">arXiv:2210.00288</a> <span> [<a href="https://arxiv.org/pdf/2210.00288">pdf</a>, <a href="https://arxiv.org/ps/2210.00288">ps</a>, <a href="https://arxiv.org/format/2210.00288">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219498824500038">10.1142/S0219498824500038 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum Supergroup $U_{r,s}(osp(1,2))$, Scasimir Operators, and Dickson polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+F">Fu Liu</a>, <a href="/search/math?searchtype=author&query=Hu%2C+N">Naihong Hu</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.00288v1-abstract-short" style="display: inline;"> We study the center of the two-parameter quantum supergroup $U_{r,s}(osp(1,2))$ using the Dickson polynomial. We show that the Scasimir operator is completely determined by the $q$-deformed Chebychev polynomial, generalizing an earlier work of Arnaudon and Bauer. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.00288v1-abstract-full" style="display: none;"> We study the center of the two-parameter quantum supergroup $U_{r,s}(osp(1,2))$ using the Dickson polynomial. We show that the Scasimir operator is completely determined by the $q$-deformed Chebychev polynomial, generalizing an earlier work of Arnaudon and Bauer. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.00288v1-abstract-full').style.display = 'none'; document.getElementById('2210.00288v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B37; 20G42; 33C47 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra Applications (2024), 2450003 (18 pages) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.00767">arXiv:2209.00767</a> <span> [<a href="https://arxiv.org/pdf/2209.00767">pdf</a>, <a href="https://arxiv.org/ps/2209.00767">ps</a>, <a href="https://arxiv.org/format/2209.00767">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Universal symplectic/orthogonal functions and general branching rules </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jin%2C+Z">Zhihong Jin</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhijun Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Danxia Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.00767v3-abstract-short" style="display: inline;"> In this paper, we first introduce a family of universal symplectic functions $sp_位(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_位(\mathbf{x}^{\pm})$, odd symplectic characters $sp_位(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_位(\mathbf{z})$ and intermediate symplectic characters as subfamilies. We then realize the universal symplectic functions by vertex… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.00767v3-abstract-full').style.display = 'inline'; document.getElementById('2209.00767v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.00767v3-abstract-full" style="display: none;"> In this paper, we first introduce a family of universal symplectic functions $sp_位(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_位(\mathbf{x}^{\pm})$, odd symplectic characters $sp_位(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_位(\mathbf{z})$ and intermediate symplectic characters as subfamilies. We then realize the universal symplectic functions by vertex operators, which naturally lead to their skew versions, and show that $sp_位(\mathbf{x}^{\pm};\mathbf{z})$ obey the general branching rules. This also gives the Gelfand-Tsetlin representations of odd symplectic characters and a transition formula between odd symplectic characters and symplectic Schur functions. Secondly we introduce a family of universal orthogonal functions $o_位(\mathbf{x}^{\pm};\mathbf{z})$ and their skew versions in a similar manner, and we provide their vertex operator realizations and obtain transition formulas and the branching rule. The universal orthogonal functions $o_位(\mathbf{x}^{\pm};\mathbf{z})$ generalize orthogonal Schur functions $o_位(\mathbf{x}^{\pm})$, odd orthogonal Schur functions $so_位(\mathbf{x}^{\pm})$, universal orthogonal characters $o_位(\mathbf{z})$ as well as intermediate orthogonal characters. Thirdly, we give vertex operator realizations for the $CB$-interpolating Schur functions $s^{CB}_位(x;尾)$ introduced by Bisi and Zygouras (Adv. Math., 2022) and the $DB$-interpolating Schur functions $s^{DB}_位(x;尾)$ interpolating between characters of type $D$ and $B$. As an application, we show $s^{CB}_位(x;尾)$ are equal to the orthosymplectic Schur polynomials $spo_位(x/尾)$, thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.00767v3-abstract-full').style.display = 'none'; document.getElementById('2209.00767v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21pp. Revised and updated version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E05; Secondary: 17B37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.05526">arXiv:2208.05526</a> <span> [<a href="https://arxiv.org/pdf/2208.05526">pdf</a>, <a href="https://arxiv.org/format/2208.05526">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3842/SIGMA.2024.041">10.3842/SIGMA.2024.041 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Skew Symplectic and Orthogonal Schur Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhijun Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Danxia Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.05526v3-abstract-short" style="display: inline;"> Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric funct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.05526v3-abstract-full').style.display = 'inline'; document.getElementById('2208.05526v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.05526v3-abstract-full" style="display: none;"> Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by Koike and Terada and satisfy the general branching rules. Furthermore, we derive the Jacobi-Trudi identities and Gelfand-Tsetlin patterns for these symmetric functions. Additionally, the vertex operator method yields their Cauchy-type identities. This demonstrates that vertex operator representations serve not only as a tool for studying symmetric functions but also offers unified realizations for skew Schur functions of types A, C, and D. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.05526v3-abstract-full').style.display = 'none'; document.getElementById('2208.05526v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIGMA 20 (2024), 041, 23 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.04775">arXiv:2208.04775</a> <span> [<a href="https://arxiv.org/pdf/2208.04775">pdf</a>, <a href="https://arxiv.org/ps/2208.04775">ps</a>, <a href="https://arxiv.org/format/2208.04775">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.aim.2024.109569">10.1016/j.aim.2024.109569 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Minor identities for Sklyanin determinants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.04775v4-abstract-short" style="display: inline;"> We explore the invariant theory of quantum symmetric spaces of orthogonal and symplectic types by employing R-matrix techniques. Our focus involves establishing connections among the quantum determinant, Sklyanin determinants associated with the orthogonal and symplectic cases, and the quantum Pfaffians over the symplectic quantum space. Drawing inspiration from twisted Yangians, we not only demon… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.04775v4-abstract-full').style.display = 'inline'; document.getElementById('2208.04775v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.04775v4-abstract-full" style="display: none;"> We explore the invariant theory of quantum symmetric spaces of orthogonal and symplectic types by employing R-matrix techniques. Our focus involves establishing connections among the quantum determinant, Sklyanin determinants associated with the orthogonal and symplectic cases, and the quantum Pfaffians over the symplectic quantum space. Drawing inspiration from twisted Yangians, we not only demonstrate but also extend the applicability of q-Jacobi identities, q-Cayley's complementary identities, q-Sylvester identities, and Muir's theorem to Sklyanin minors in both orthogonal and symplectic types, along with q-Pfaffian analogs in the symplectic scenario. Furthermore, we present expressions for Sklyanin determinants and quantum Pfaffians in terms of quasideterminants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.04775v4-abstract-full').style.display = 'none'; document.getElementById('2208.04775v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">32 pages; Final version for publication</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 17B37; Secondary: 58A17; 15A75; 15B33; 15A15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv. Math. (2024) 109569 (30pp) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.00303">arXiv:2208.00303</a> <span> [<a href="https://arxiv.org/pdf/2208.00303">pdf</a>, <a href="https://arxiv.org/format/2208.00303">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.12958/adm2006">10.12958/adm2006 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On Smith normal forms of $q$-Varchenko matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Boulware%2C+N">Naomi Boulware</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Misra%2C+K+C">Kailash C. Misra</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.00303v1-abstract-short" style="display: inline;"> In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the hyperplane arrangement for the regular $n$-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the $q$-Varchenko matrix as… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.00303v1-abstract-full').style.display = 'inline'; document.getElementById('2208.00303v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.00303v1-abstract-full" style="display: none;"> In this paper, we investigate $q$-Varchenko matrices for some hyperplane arrangements with symmetry in two and three dimensions, and prove that they have a Smith normal form over $\mathbb Z[q]$. In particular, we examine the hyperplane arrangement for the regular $n$-gon in the plane and the dihedral model in the space and Platonic polyhedra. In each case, we prove that the $q$-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over $\mathbb Z[q]$ and realize their congruent transformation matrices over $\mathbb Z[q]$ as well. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.00303v1-abstract-full').style.display = 'none'; document.getElementById('2208.00303v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 8 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algebra and Discrete Math. 34:2 (2022), 187-222 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.01712">arXiv:2207.01712</a> <span> [<a href="https://arxiv.org/pdf/2207.01712">pdf</a>, <a href="https://arxiv.org/ps/2207.01712">ps</a>, <a href="https://arxiv.org/format/2207.01712">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11425-022-2142-9">10.1007/s11425-022-2142-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Center of the Yangian double in type A </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fan%2C+Y">Yang Fan</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.01712v2-abstract-short" style="display: inline;"> We prove the R-matrix and Drinfeld presentations of the Yangian double in type A are isomorphic. The central elements of the completed Yangian double in type A at the critical level are constructed. The images of these elements under a Harish-Chandra-type homomorphism are calculated by applying a version of the Poincar茅-Birkhoff-Witt theorem for the R-matrix presentation. These images coincide wit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01712v2-abstract-full').style.display = 'inline'; document.getElementById('2207.01712v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.01712v2-abstract-full" style="display: none;"> We prove the R-matrix and Drinfeld presentations of the Yangian double in type A are isomorphic. The central elements of the completed Yangian double in type A at the critical level are constructed. The images of these elements under a Harish-Chandra-type homomorphism are calculated by applying a version of the Poincar茅-Birkhoff-Witt theorem for the R-matrix presentation. These images coincide with the eigenvalues of the central elements in the Wakimoto modules. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01712v2-abstract-full').style.display = 'none'; document.getElementById('2207.01712v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages, no figure. Final version accepted by SCIENCE CHINA Mathematics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Sci. China Math. 67 (2024), 1957-1988 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.11175">arXiv:2202.11175</a> <span> [<a href="https://arxiv.org/pdf/2202.11175">pdf</a>, <a href="https://arxiv.org/ps/2202.11175">ps</a>, <a href="https://arxiv.org/format/2202.11175">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.aam.2023.102630">10.1016/j.aam.2023.102630 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A note on Cauchy's formula </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhijun Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.11175v3-abstract-short" style="display: inline;"> We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{渭\subseteq [K\times N]}(-1)^{|渭|}s_渭\{x\}s_{渭'}\{y\}. \end{align*} As an application of the interpretation, we obtain an expansion of $\prod^\infty_{i=1}(1-q^i)^{i-1}$ in terms of half plane partitions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.11175v3-abstract-full" style="display: none;"> We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{渭\subseteq [K\times N]}(-1)^{|渭|}s_渭\{x\}s_{渭'}\{y\}. \end{align*} As an application of the interpretation, we obtain an expansion of $\prod^\infty_{i=1}(1-q^i)^{i-1}$ in terms of half plane partitions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.11175v3-abstract-full').style.display = 'none'; document.getElementById('2202.11175v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, no figure. Final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv. Appl. Math. 153 (2024), 102630 (14pp.) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.03403">arXiv:2202.03403</a> <span> [<a href="https://arxiv.org/pdf/2202.03403">pdf</a>, <a href="https://arxiv.org/ps/2202.03403">ps</a>, <a href="https://arxiv.org/format/2202.03403">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11464-021-0434-7">10.1007/s11464-021-0434-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> R-Matrix presentation of quantum affine algebra in type $A_{2n-1}^{(2)}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">Xia Zhang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.03403v1-abstract-short" style="display: inline;"> In this paper, we give an RTT presentation of the twisted quantum affine algebra of type $A_{2n-1}^{(2)}$ and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first such presentation for twisted quantum affine algebras with nontrivial central element. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.03403v1-abstract-full" style="display: none;"> In this paper, we give an RTT presentation of the twisted quantum affine algebra of type $A_{2n-1}^{(2)}$ and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first such presentation for twisted quantum affine algebras with nontrivial central element. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.03403v1-abstract-full').style.display = 'none'; document.getElementById('2202.03403v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">42pp. arXiv admin note: text overlap with arXiv:1903.00204</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Front. Math. 18 (2023), No.3, 513-564 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.06067">arXiv:2104.06067</a> <span> [<a href="https://arxiv.org/pdf/2104.06067">pdf</a>, <a href="https://arxiv.org/ps/2104.06067">ps</a>, <a href="https://arxiv.org/format/2104.06067">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2022.01.020">10.1016/j.jalgebra.2022.01.020 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On irreducible characters of the Iwahori-Hecke algebra in type $A$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.06067v1-abstract-short" style="display: inline;"> We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.06067v1-abstract-full').style.display = 'inline'; document.getElementById('2104.06067v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.06067v1-abstract-full" style="display: none;"> We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type $A$. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.06067v1-abstract-full').style.display = 'none'; document.getElementById('2104.06067v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 598 (2022) 24-47 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.05223">arXiv:2104.05223</a> <span> [<a href="https://arxiv.org/pdf/2104.05223">pdf</a>, <a href="https://arxiv.org/ps/2104.05223">ps</a>, <a href="https://arxiv.org/format/2104.05223">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2021.10.030">10.1016/j.jalgebra.2021.10.030 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lattice structure of modular vertex algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+H">Haihua Huang</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.05223v4-abstract-short" style="display: inline;"> In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe ana… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.05223v4-abstract-full').style.display = 'inline'; document.getElementById('2104.05223v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.05223v4-abstract-full" style="display: none;"> In this paper we study the integral form of the lattice vertex algebra $V_L$. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant $\mathbb Z$-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.05223v4-abstract-full').style.display = 'none'; document.getElementById('2104.05223v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages; final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 592 (2022), 1--17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.04411">arXiv:2104.04411</a> <span> [<a href="https://arxiv.org/pdf/2104.04411">pdf</a>, <a href="https://arxiv.org/ps/2104.04411">ps</a>, <a href="https://arxiv.org/format/2104.04411">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jpaa.2022.107032">10.1016/j.jpaa.2022.107032 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Green polynomials via vertex operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+N">Ning Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.04411v2-abstract-short" style="display: inline;"> An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.04411v2-abstract-full').style.display = 'inline'; document.getElementById('2104.04411v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.04411v2-abstract-full" style="display: none;"> An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.04411v2-abstract-full').style.display = 'none'; document.getElementById('2104.04411v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages; updated version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Pure Appl. Algebra 226 (2022), 107032, 17pp </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.01464">arXiv:2101.01464</a> <span> [<a href="https://arxiv.org/pdf/2101.01464">pdf</a>, <a href="https://arxiv.org/ps/2101.01464">ps</a>, <a href="https://arxiv.org/format/2101.01464">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219199722500675">10.1142/S0219199722500675 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Deforming vertex algebras by vertex bialgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Kong%2C+F">Fei Kong</a>, <a href="/search/math?searchtype=author&query=Li%2C+H">Haisheng Li</a>, <a href="/search/math?searchtype=author&query=Tan%2C+S">Shaobin Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.01464v1-abstract-short" style="display: inline;"> This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-modul… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.01464v1-abstract-full').style.display = 'inline'; document.getElementById('2101.01464v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.01464v1-abstract-full" style="display: none;"> This is a continuation of a previous study initiated by one of us on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ and give a construction of deformations of vertex algebras with a right $H$-comodule nonlocal vertex algebra structure and a compatible $H$-module nonlocal vertex algebra structure. We also give a construction of $蠁$-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.01464v1-abstract-full').style.display = 'none'; document.getElementById('2101.01464v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Comm. Contemp. Math. 26 (2024) No.1, 2250067 (52 pages) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.07847">arXiv:2008.07847</a> <span> [<a href="https://arxiv.org/pdf/2008.07847">pdf</a>, <a href="https://arxiv.org/format/2008.07847">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3842/SIGMA.2020.145">10.3842/SIGMA.2020.145 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Representations of Quantum Affine Algebras in their $R$-Matrix Realization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Liu%2C+M">Ming Liu</a>, <a href="/search/math?searchtype=author&query=Molev%2C+A">Alexander Molev</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.07847v2-abstract-short" style="display: inline;"> We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.07847v2-abstract-full').style.display = 'inline'; document.getElementById('2008.07847v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.07847v2-abstract-full" style="display: none;"> We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.07847v2-abstract-full').style.display = 'none'; document.getElementById('2008.07847v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">includes a review of arXiv:1705.08155, arXiv:1903.00204 and arXiv:1911.03496</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIGMA 16 (2020), 145, 25 pages </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.05982">arXiv:2008.05982</a> <span> [<a href="https://arxiv.org/pdf/2008.05982">pdf</a>, <a href="https://arxiv.org/ps/2008.05982">ps</a>, <a href="https://arxiv.org/format/2008.05982">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2020.11.013">10.1016/j.jalgebra.2020.11.013 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> $(G,蠂_蠁)$-equivariant $蠁$-coordinated quasi modules for nonlocal vertex algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Kong%2C+F">Fei Kong</a>, <a href="/search/math?searchtype=author&query=Li%2C+H">Haisheng Li</a>, <a href="/search/math?searchtype=author&query=Tan%2C+S">Shaobin Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.05982v1-abstract-short" style="display: inline;"> In this paper, we study $(G,蠂_蠁)$-equivariant $蠁$-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their $(G,蠂_蠁)$-equivariant $蠁$-coordinated quasi modules. As an application, we also construct (equivariant) $蠁$-coordinat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.05982v1-abstract-full').style.display = 'inline'; document.getElementById('2008.05982v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.05982v1-abstract-full" style="display: none;"> In this paper, we study $(G,蠂_蠁)$-equivariant $蠁$-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their $(G,蠂_蠁)$-equivariant $蠁$-coordinated quasi modules. As an application, we also construct (equivariant) $蠁$-coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.05982v1-abstract-full').style.display = 'none'; document.getElementById('2008.05982v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Algebra 570 (2021), 24-74 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.15791">arXiv:2007.15791</a> <span> [<a href="https://arxiv.org/pdf/2007.15791">pdf</a>, <a href="https://arxiv.org/ps/2007.15791">ps</a>, <a href="https://arxiv.org/format/2007.15791">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S1005386722000074">10.1142/S1005386722000074 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Level $-1/2$ realization of quantum N-toroidal algebras in type $C_n$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Q">Qianbao Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.15791v1-abstract-short" style="display: inline;"> We construct a level $-\frac{1}{2}$ vertex representation of the quantum N-toroidal algebra for type $C_n$, which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type $C_n$ as a by-product. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.15791v1-abstract-full" style="display: none;"> We construct a level $-\frac{1}{2}$ vertex representation of the quantum N-toroidal algebra for type $C_n$, which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type $C_n$ as a by-product. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.15791v1-abstract-full').style.display = 'none'; document.getElementById('2007.15791v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B37; 17B67 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Alg. Colloq. 29:1 (2022) 79-98 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.11441">arXiv:2007.11441</a> <span> [<a href="https://arxiv.org/pdf/2007.11441">pdf</a>, <a href="https://arxiv.org/ps/2007.11441">ps</a>, <a href="https://arxiv.org/format/2007.11441">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1080/00927872.2022.2154783">10.1080/00927872.2022.2154783 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> $\mathcal{O}$-operators and related structures on Leibniz algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sun%2C+Q">Qinxiu Sun</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.11441v3-abstract-short" style="display: inline;"> An $\mathcal{O}$-operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to $\mathcal{O}$-operators on Leibniz algebras and introduce (dual) $\mathcal{O}$N-structures on Leibniz algebras associated to their representations. It is proved that $\mathcal{O}$-operators and dual $\mathcal{O}$N-structures generate each other un… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.11441v3-abstract-full').style.display = 'inline'; document.getElementById('2007.11441v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.11441v3-abstract-full" style="display: none;"> An $\mathcal{O}$-operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to $\mathcal{O}$-operators on Leibniz algebras and introduce (dual) $\mathcal{O}$N-structures on Leibniz algebras associated to their representations. It is proved that $\mathcal{O}$-operators and dual $\mathcal{O}$N-structures generate each other under certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual $\mathcal{O}$N-structure. Finally, $r-n$ structures, RBN-structures and $\mathcal{B}N$-structures on Leibniz algebras are thoroughly studied and their interdependent relations are also studied. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.11441v3-abstract-full').style.display = 'none'; document.getElementById('2007.11441v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, new title</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 13D03; 16T10; 16B38 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Comm. Algebra 51:5 (2023), 2199-2216 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.14783">arXiv:2006.14783</a> <span> [<a href="https://arxiv.org/pdf/2006.14783">pdf</a>, <a href="https://arxiv.org/ps/2006.14783">ps</a>, <a href="https://arxiv.org/format/2006.14783">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1090/tran/8706">10.1090/tran/8706 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Twisted quantum affinizations and quantization of extended affine Lie algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+F">Fulin Chen</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Kong%2C+F">Fei Kong</a>, <a href="/search/math?searchtype=author&query=Tan%2C+S">Shaobin Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.14783v2-abstract-short" style="display: inline;"> In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $渭$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $渭$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ is Drinfeld… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.14783v2-abstract-full').style.display = 'inline'; document.getElementById('2006.14783v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.14783v2-abstract-full" style="display: none;"> In this paper, for an arbitrary Kac-Moody Lie algebra $\mathfrak g$ and a diagram automorphism $渭$ of $\mathfrak g$ satisfying certain natural linking conditions, we introduce and study a $渭$-twisted quantum affinization algebra $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ of $\mathfrak g$. When $\mathfrak g$ is of finite type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ is Drinfeld's current algebra realization of the twisted quantum affine algebra. When $渭=\mathrm{id}$ and $\mathfrak g$ in affine type, $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$. Second, we give a simple characterization of the affine quantum Serre relations on restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$-modules in terms of "normal order products". Third, we prove that the category of restricted $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$. Last, we study the classical limit of $\mathcal U_\hbar\left(\hat{\mathfrak g}_渭\right)$ and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the $\hbar$-deformation of all nullity $2$ extended affine Lie algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.14783v2-abstract-full').style.display = 'none'; document.getElementById('2006.14783v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">66 pages. Final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Trans. Amer. Math. Soc. 376 (2) (2023), 969-1039 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.14558">arXiv:2006.14558</a> <span> [<a href="https://arxiv.org/pdf/2006.14558">pdf</a>, <a href="https://arxiv.org/ps/2006.14558">ps</a>, <a href="https://arxiv.org/format/2006.14558">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jpaa.2021.106814">10.1016/j.jpaa.2021.106814 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On quantum toroidal algebra of type $A_1$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+F">Fulin Chen</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Kong%2C+F">Fei Kong</a>, <a href="/search/math?searchtype=author&query=Tan%2C+S">Shaobin Tan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.14558v1-abstract-short" style="display: inline;"> In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.14558v1-abstract-full" style="display: none;"> In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.14558v1-abstract-full').style.display = 'none'; document.getElementById('2006.14558v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Pure Applied Algebra (2022), no.1, 108614 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.02025">arXiv:2006.02025</a> <span> [<a href="https://arxiv.org/pdf/2006.02025">pdf</a>, <a href="https://arxiv.org/ps/2006.02025">ps</a>, <a href="https://arxiv.org/format/2006.02025">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.37236/8091">10.37236/8091 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Deformation of Cayley's hyperdeterminants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cai%2C+T+W">Tommy Wuxing Cai</a>, <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.02025v2-abstract-short" style="display: inline;"> We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.02025v2-abstract-full" style="display: none;"> We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing Matsumoto's formula for Jack functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.02025v2-abstract-full').style.display = 'none'; document.getElementById('2006.02025v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 0 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 05E05; Secondary: 17B69; 05E10 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Elec. J. Combin. 27(2) (2020) P2.50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.00597">arXiv:2006.00597</a> <span> [<a href="https://arxiv.org/pdf/2006.00597">pdf</a>, <a href="https://arxiv.org/ps/2006.00597">ps</a>, <a href="https://arxiv.org/format/2006.00597">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219498821501851">10.1142/S0219498821501851 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Vertex representations of quantum N-toroidal algebras for type C </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jing%2C+N">Naihuan Jing</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Z">Zhucheng Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Honglian Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.00597v2-abstract-short" style="display: inline;"> Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.00597v2-abstract-full" style="display: none;"> Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.00597v2-abstract-full').style.display = 'none'; document.getElementById('2006.00597v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, no figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B37; 17B67 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Alg. Appl. 20 (2021), no. 10, 2150185 (18 pages) </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Jing%2C+N&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Jing%2C+N&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Jing%2C+N&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Jing%2C+N&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Jing%2C+N&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> </ul> </nav> <div class="is-hidden-tablet">  <span class="help" 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