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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.19505">arXiv:2502.19505</a> <span> [<a href="https://arxiv.org/pdf/2502.19505">pdf</a>, <a href="https://arxiv.org/ps/2502.19505">ps</a>, <a href="https://arxiv.org/format/2502.19505">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> </div> </div> <p class="title is-5 mathjax"> $K$-type multiplicities in degenerate principal series via Howe duality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Colarusso%2C+M">Mark Colarusso</a>, <a href="/search/math?searchtype=author&query=Erickson%2C+W+Q">William Q. Erickson</a>, <a href="/search/math?searchtype=author&query=Frohmader%2C+A">Andrew Frohmader</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="2502.19505v1-abstract-short" style="display: inline;"> Let $K$ be one of the complex classical groups ${\rm O}_k$, ${\rm GL}_k$, or ${\rm Sp}_{2k}$. Let $M \subseteq K$ be the block diagonal embedding ${\rm O}_{k_1} \times \cdots \times {\rm O}_{k_r}$ or ${\rm GL}_{k_1} \times \cdots \times {\rm GL}_{k_r}$ or ${\rm Sp}_{2k_1} \times \cdots \times {\rm Sp}_{2k_r}$, respectively. By using Howe duality and seesaw reciprocity as a unified conceptual frame… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.19505v1-abstract-full').style.display = 'inline'; document.getElementById('2502.19505v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.19505v1-abstract-full" style="display: none;"> Let $K$ be one of the complex classical groups ${\rm O}_k$, ${\rm GL}_k$, or ${\rm Sp}_{2k}$. Let $M \subseteq K$ be the block diagonal embedding ${\rm O}_{k_1} \times \cdots \times {\rm O}_{k_r}$ or ${\rm GL}_{k_1} \times \cdots \times {\rm GL}_{k_r}$ or ${\rm Sp}_{2k_1} \times \cdots \times {\rm Sp}_{2k_r}$, respectively. By using Howe duality and seesaw reciprocity as a unified conceptual framework, we prove a formula for the branching multiplicities from $K$ to $M$ which is expressed as a sum of generalized Littlewood-Richardson coefficients, valid within a certain stable range. By viewing $K$ as the complexification of the maximal compact subgroup $K_{\mathbb{R}}$ of the real group $G_{\mathbb{R}} = {\rm GL}(k,\mathbb{R})$, ${\rm GL}(k, \mathbb{C})$, or ${\rm GL}(k,\mathbb{H})$, respectively, one can interpret our branching multiplicities as $K_{\mathbb{R}}$-type multiplicities in degenerate principal series representations of $G_{\mathbb{R}}$. Upon specializing to the minimal $M$, where $k_1 = \cdots = k_r = 1$, we establish a fully general tableau-theoretic interpretation of the branching multiplicities, corresponding to the $K_{\mathbb{R}}$-type multiplicities in the principal series. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.19505v1-abstract-full').style.display = 'none'; document.getElementById('2502.19505v1-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, 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">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G05 (Primary) 05E10; 17B10 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.08708">arXiv:2205.08708</a> <span> [<a href="https://arxiv.org/pdf/2205.08708">pdf</a>, <a href="https://arxiv.org/format/2205.08708">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> </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/10.4153/S0008414X23000780">10.4153/S0008414X23000780 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Graphical methods and rings of invariants on the symmetric algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bourn%2C+R">Rebecca Bourn</a>, <a href="/search/math?searchtype=author&query=Erickson%2C+W+Q">William Q. Erickson</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="2205.08708v3-abstract-short" style="display: inline;"> Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of $G$-invariant polynomial functions on the space $\mathcal{P}^m(V)$ of degree-$m$ homogeneous polynomial functions on $V$. In this paper, we replace $\mathcal{P}^m(V)$ with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.08708v3-abstract-full').style.display = 'inline'; document.getElementById('2205.08708v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.08708v3-abstract-full" style="display: none;"> Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of $G$-invariant polynomial functions on the space $\mathcal{P}^m(V)$ of degree-$m$ homogeneous polynomial functions on $V$. In this paper, we replace $\mathcal{P}^m(V)$ with the full polynomial algebra $\mathcal{P}(V)$. As a result, the invariant ring is no longer finitely generated. Hence instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when $G$ is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of $G$ is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal{P}(V)$. We conclude with examples using our graphical notation, several of which recover classical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.08708v3-abstract-full').style.display = 'none'; document.getElementById('2205.08708v3-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 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">26 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E10 (Primary); 16W22 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Can. J. Math.-J. Can. Math. 76 (2024) 2173-2198 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.06928">arXiv:2012.06928</a> <span> [<a href="https://arxiv.org/pdf/2012.06928">pdf</a>, <a href="https://arxiv.org/ps/2012.06928">ps</a>, <a href="https://arxiv.org/format/2012.06928">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> </div> </div> <p class="title is-5 mathjax"> Contingency tables and the generalized Littlewood-Richardson coefficients </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Colarusso%2C+M">Mark Colarusso</a>, <a href="/search/math?searchtype=author&query=Erickson%2C+W+Q">William Q. Erickson</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="2012.06928v1-abstract-short" style="display: inline;"> The Littlewood-Richardson coefficients $c^位_{渭谓}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^位_n$ in the tensor product of polynomial representations $F^渭_n\otimes F^谓_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical continge… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.06928v1-abstract-full').style.display = 'inline'; document.getElementById('2012.06928v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.06928v1-abstract-full" style="display: none;"> The Littlewood-Richardson coefficients $c^位_{渭谓}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^位_n$ in the tensor product of polynomial representations $F^渭_n\otimes F^谓_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.06928v1-abstract-full').style.display = 'none'; document.getElementById('2012.06928v1-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 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">15 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G20 (Primary); 17B10; 05E10 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.03673">arXiv:1903.03673</a> <span> [<a href="https://arxiv.org/pdf/1903.03673">pdf</a>, <a href="https://arxiv.org/ps/1903.03673">ps</a>, <a href="https://arxiv.org/format/1903.03673">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/10.2140/astat.2020.11.53">10.2140/astat.2020.11.53 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Expected value of the one-dimensional Earth Mover's Distance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bourn%2C+R">Rebecca Bourn</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="1903.03673v2-abstract-short" style="display: inline;"> From a combinatorial point of view, we consider the Earth Mover's Distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set [n] = {1,...,n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy metho… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.03673v2-abstract-full').style.display = 'inline'; document.getElementById('1903.03673v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.03673v2-abstract-full" style="display: none;"> From a combinatorial point of view, we consider the Earth Mover's Distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set [n] = {1,...,n} be regarded as a metric space by restricting the usual Euclidean distance on the real numbers. The EMD is defined on ordered pairs of probability distributions on [n]. We provide an easy method to compute a generating function encoding the values of EMD in its coefficients, which is related to the Segre embedding from projective algebraic geometry. As an application we use the generating function to compute the expected value of EMD in this one-dimensional case. The EMD is then used in clustering analysis for a specific data set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.03673v2-abstract-full').style.display = 'none'; document.getElementById('1903.03673v2-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 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </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">To appear in the Journal of Algebraic Statistics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E40; 62H30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Alg. Stat. 11 (2020) 53-78 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.06211">arXiv:1812.06211</a> <span> [<a href="https://arxiv.org/pdf/1812.06211">pdf</a>, <a href="https://arxiv.org/format/1812.06211">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> </div> </div> <p class="title is-5 mathjax"> Branching from the General Linear Group to the Symmetric Group and the Principal Embedding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Heaton%2C+A">Alexander Heaton</a>, <a href="/search/math?searchtype=author&query=Sriwongsa%2C+S">Songpon Sriwongsa</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="1812.06211v2-abstract-short" style="display: inline;"> Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n representation, V, there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.06211v2-abstract-full').style.display = 'inline'; document.getElementById('1812.06211v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.06211v2-abstract-full" style="display: none;"> Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n representation, V, there exists an irreducible S-representation embedding in V with dimension at most b(n). In a 2017 paper (joint with Hassan Lhou), they prove that b(n)=n is the sharpest possible bound, and also address embeddings other than the principal one. These results concerning embeddings may by interpreted as statements about plethysm. Then, a well known result about these plethysms can be interpreted as a "branching rule". Specifically, a (finite dimensional) representation of GL(n,C) will decompose into irreducible representations of the symmetric group when it is restricted to the subgroup consisting of permutation matrices. The question of which irreducible representations of the symmetric group occur with positive multiplicity is the topic of this paper, applying the previous work of Lhou, Zuckerman, and the third author. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.06211v2-abstract-full').style.display = 'none'; document.getElementById('1812.06211v2-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> 9 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </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">Revisions based on referee comments</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20C30; 05E10; 22E46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.04935">arXiv:1603.04935</a> <span> [<a href="https://arxiv.org/pdf/1603.04935">pdf</a>, <a href="https://arxiv.org/ps/1603.04935">ps</a>, <a href="https://arxiv.org/format/1603.04935">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> </div> </div> <p class="title is-5 mathjax"> Lowest sl(2)-types in sl(n)-representations with respect to a principal embedding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lhou%2C+H">Hassan Lhou</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="1603.04935v3-abstract-short" style="display: inline;"> Fix n>2. Let s be a principally embedded sl(2)-subalgebra in sl(n). A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional sl(n)-representation, V, there exists an irreducible s-representation embedding in V with dimension at most b(n). We prove that b(n)=n is the sharpest possible bound. We also add… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.04935v3-abstract-full').style.display = 'inline'; document.getElementById('1603.04935v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.04935v3-abstract-full" style="display: none;"> Fix n>2. Let s be a principally embedded sl(2)-subalgebra in sl(n). A special case of results of the second author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite dimensional sl(n)-representation, V, there exists an irreducible s-representation embedding in V with dimension at most b(n). We prove that b(n)=n is the sharpest possible bound. We also address embeddings other than the principal one. The exposition involves an application of the Cartan--Helgason theorem, Pieri rules, Hermite reciprocity, and a calculation in the "branching algebra" introduced by Roger Howe, Eng-Chye Tan, and the second author. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.04935v3-abstract-full').style.display = 'none'; document.getElementById('1603.04935v3-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> 23 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </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. Possible newer version at http://www.uwm.edu/~jw/PAPERS/LW.pdf</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E10; 22E46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1206.0404">arXiv:1206.0404</a> <span> [<a href="https://arxiv.org/pdf/1206.0404">pdf</a>, <a href="https://arxiv.org/format/1206.0404">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> </div> </div> <p class="title is-5 mathjax"> Sums of squares of the Littlewood-Richardson coefficients and GL(n)-harmonic polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Harris%2C+P+E">Pamela E. Harris</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="1206.0404v2-abstract-short" style="display: inline;"> We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coeff… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.0404v2-abstract-full').style.display = 'inline'; document.getElementById('1206.0404v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1206.0404v2-abstract-full" style="display: none;"> We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coefficients. A "principal specialization" of the gradation is then related to the Hilbert series of the $\K$-invariant subring in the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1206.0404v2-abstract-full').style.display = 'none'; document.getElementById('1206.0404v2-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 April, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 June, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2012. </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">Possible updated version located at: https://pantherfile.uwm.edu/jw/www/PAPERS/Harris_Willenbring.pdf</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G05; 22E46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0911.0222">arXiv:0911.0222</a> <span> [<a href="https://arxiv.org/pdf/0911.0222">pdf</a>, <a href="https://arxiv.org/ps/0911.0222">ps</a>, <a href="https://arxiv.org/format/0911.0222">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> </div> </div> <p class="title is-5 mathjax"> The measurement of quantum entanglement and enumeration of graph coverings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hero%2C+M+W">Michael W. Hero</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</a>, <a href="/search/math?searchtype=author&query=Williams%2C+L+K">Lauren Kelly Williams</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="0911.0222v3-abstract-short" style="display: inline;"> We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entang… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0911.0222v3-abstract-full').style.display = 'inline'; document.getElementById('0911.0222v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0911.0222v3-abstract-full" style="display: none;"> We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entangled and unentangled states, and are therefore viewed as measurements of quantum entanglement. When the ranks of the unitary groups are large, we provide a graph theoretic interpretation for the dimension of the invariants of a fixed degree. We also exhibit a bijection between isomorphism classes of finite coverings of connected simple graphs and a basis for the space of invariants. The graph coverings are related to branched coverings of surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0911.0222v3-abstract-full').style.display = 'none'; document.getElementById('0911.0222v3-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 October, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 November, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2009. </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">Version to appear in the AMS Contemporary Mathematics Series</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 22E70; 81P15; 05C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0612266">arXiv:math/0612266</a> <span> [<a href="https://arxiv.org/pdf/math/0612266">pdf</a>, <a href="https://arxiv.org/ps/math/0612266">ps</a>, <a href="https://arxiv.org/format/math/0612266">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> </div> </div> <p class="title is-5 mathjax"> A generating function for Blattner's formula </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</a>, <a href="/search/math?searchtype=author&query=Zuckerman%2C+G+J">Gregg J. Zuckerman</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="math/0612266v2-abstract-short" style="display: inline;"> Let G be a connected, semisimple Lie group with finite center and let K be a maximal compact subgroup. We investigate a method to compute multiplicities of K-types in the discrete series using a rational expression for a generating function obtained from Blattner's formula. This expression involves a product with a character of an irreducible finite dimensional representation of K and is valid f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0612266v2-abstract-full').style.display = 'inline'; document.getElementById('math/0612266v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0612266v2-abstract-full" style="display: none;"> Let G be a connected, semisimple Lie group with finite center and let K be a maximal compact subgroup. We investigate a method to compute multiplicities of K-types in the discrete series using a rational expression for a generating function obtained from Blattner's formula. This expression involves a product with a character of an irreducible finite dimensional representation of K and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattner's formula, and a positivity result for certain low rank examples. We consider in detail the situation for G of type split G_2. The motivation for this work came from an attempt to understand pictures coming from Blattner's formula, some of which we include in the paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0612266v2-abstract-full').style.display = 'none'; document.getElementById('math/0612266v2-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> 17 December, 2006; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 December, 2006; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2006. </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">12 pages, 4 figures, added section on a positivity result</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0510649">arXiv:math/0510649</a> <span> [<a href="https://arxiv.org/pdf/math/0510649">pdf</a>, <a href="https://arxiv.org/ps/math/0510649">ps</a>, <a href="https://arxiv.org/format/math/0510649">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> </div> </div> <p class="title is-5 mathjax"> Stable Hilbert series of $\mathcal S(\mathfrak g)^K$ for classical groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="math/0510649v2-abstract-short" style="display: inline;"> Given a classical symmetric pair, $(G,K)$, with $\mathfrak g = Lie(G)$, we provide descriptions of the Hilbert series of the algebra of $K$-invariant vectors in the associated graded algebra of $\mathcal U(\mathfrak g)$ viewed as a $K$-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0510649v2-abstract-full').style.display = 'inline'; document.getElementById('math/0510649v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0510649v2-abstract-full" style="display: none;"> Given a classical symmetric pair, $(G,K)$, with $\mathfrak g = Lie(G)$, we provide descriptions of the Hilbert series of the algebra of $K$-invariant vectors in the associated graded algebra of $\mathcal U(\mathfrak g)$ viewed as a $K$-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note that the stable Hilbert series of one symmetric pair often coincides with others. Also, for the case of the real form $U(p,q)$ we derive a closed expression for the Hilbert series when $\min(p,q) \to \infty$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0510649v2-abstract-full').style.display = 'none'; document.getElementById('math/0510649v2-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 November, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 October, 2005; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2005. </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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0409561">arXiv:math/0409561</a> <span> [<a href="https://arxiv.org/pdf/math/0409561">pdf</a>, <a href="https://arxiv.org/ps/math/0409561">ps</a>, <a href="https://arxiv.org/format/math/0409561">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> </div> </div> <p class="title is-5 mathjax"> Invariant Differential Operators and FCR factors of Enveloping algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Musson%2C+I+M">Ian M. Musson</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="math/0409561v3-abstract-short" style="display: inline;"> If $\fg$ is a semisimple Lie algebra, we describe the prime factors of $\mcU(\fg)$ that have enough finite dimensional modules. The proof depends on some combinatorial facts about the Weyl group which may be of independent interest. We also determine, which finite dimensional $\mcU(\fg)$-modules are modules over a given prime factor. As an application we study finite dimensional modules over som… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0409561v3-abstract-full').style.display = 'inline'; document.getElementById('math/0409561v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0409561v3-abstract-full" style="display: none;"> If $\fg$ is a semisimple Lie algebra, we describe the prime factors of $\mcU(\fg)$ that have enough finite dimensional modules. The proof depends on some combinatorial facts about the Weyl group which may be of independent interest. We also determine, which finite dimensional $\mcU(\fg)$-modules are modules over a given prime factor. As an application we study finite dimensional modules over some rings of invariant differential operators arising from Howe duality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0409561v3-abstract-full').style.display = 'none'; document.getElementById('math/0409561v3-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> 16 March, 2007; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 September, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2004. </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, updated version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0408302">arXiv:math/0408302</a> <span> [<a href="https://arxiv.org/pdf/math/0408302">pdf</a>, <a href="https://arxiv.org/ps/math/0408302">ps</a>, <a href="https://arxiv.org/format/math/0408302">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> </div> </div> <p class="title is-5 mathjax"> Small semisimple subalgebras of semisimple Lie algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</a>, <a href="/search/math?searchtype=author&query=Zuckerman%2C+G">Gregg Zuckerman</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="math/0408302v1-abstract-short" style="display: inline;"> The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak k, \frak g)$, such that for every irreducible finite dimensional $\frak g$-module $V$, there exists an injection of $\frak k$-modules $W \to V$, where $W$ i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0408302v1-abstract-full').style.display = 'inline'; document.getElementById('math/0408302v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0408302v1-abstract-full" style="display: none;"> The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak k, \frak g)$, such that for every irreducible finite dimensional $\frak g$-module $V$, there exists an injection of $\frak k$-modules $W \to V$, where $W$ is an irreducible $\frak k$-module of dimension less than $b(\frak k, \frak g)$. This result was announced in math.RT/0310140. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0408302v1-abstract-full').style.display = 'none'; document.getElementById('math/0408302v1-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> 23 August, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2004. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 17B10; 20G05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0407468">arXiv:math/0407468</a> <span> [<a href="https://arxiv.org/pdf/math/0407468">pdf</a>, <a href="https://arxiv.org/ps/math/0407468">ps</a>, <a href="https://arxiv.org/format/math/0407468">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> </div> </div> <p class="title is-5 mathjax"> A Basis for the GL_n Tensor Product Algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Howe%2C+R+E">Roger E. Howe</a>, <a href="/search/math?searchtype=author&query=Tan%2C+E+C">Eng Chye Tan</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="math/0407468v2-abstract-short" style="display: inline;"> This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra. This construction relates directly with the combinatorial description of Littlewood-Richardson coefficients in terms of Littlewood-Richardson tableaux. Philosoph… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0407468v2-abstract-full').style.display = 'inline'; document.getElementById('math/0407468v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0407468v2-abstract-full" style="display: none;"> This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra. This construction relates directly with the combinatorial description of Littlewood-Richardson coefficients in terms of Littlewood-Richardson tableaux. Philosophically, one may view this construction as a recasting of the Littlewood-Richardson rule in the context of classical invariant theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0407468v2-abstract-full').style.display = 'none'; document.getElementById('math/0407468v2-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 November, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 July, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2004. </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">34 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Mathematics; Vol 196/2 pp 531-564 (2005) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0407467">arXiv:math/0407467</a> <span> [<a href="https://arxiv.org/pdf/math/0407467">pdf</a>, <a href="https://arxiv.org/ps/math/0407467">ps</a>, <a href="https://arxiv.org/format/math/0407467">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> </div> </div> <p class="title is-5 mathjax"> Reciprocity Algebras and Branching for Classical Symmetric Pairs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Howe%2C+R+E">Roger E. Howe</a>, <a href="/search/math?searchtype=author&query=Tan%2C+E+C">Eng Chye Tan</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="math/0407467v2-abstract-short" style="display: inline;"> We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag manifold for $G$. We give concrete descriptions of (natural subalgebras of) the branching algebra using classical invariant theory. In this context, it turns out th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0407467v2-abstract-full').style.display = 'inline'; document.getElementById('math/0407467v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0407467v2-abstract-full" style="display: none;"> We study branching laws for a classical group $G$ and a symmetric subgroup $H$. Our approach is through the {\it branching algebra}, the algebra of covariants for $H$ in the regular functions on the natural torus bundle over the flag manifold for $G$. We give concrete descriptions of (natural subalgebras of) the branching algebra using classical invariant theory. In this context, it turns out that the ten classes of classical symmetric pairs $(G,H)$ are associated in pairs, $(G,H)$ and $(H',G')$, and that the (partial) branching algebra for $(G,H)$ also describes a branching law from $H'$ to $G'$. (However, the second branching law may involve certain infinite-dimensional highest weight modules for $H'$.) To highlight the fact that these algebras describe two branching laws simultaneously, we call them {\it reciprocity algebras}. Our description of the reciprocity algebras reveals that they all are related to the tensor product algebra for $GL_n$. This relation is especially strong in the {\it stable range}. We give quite explicit descriptions of reciprocity algebras in the stable range in terms of the tensor product algebra for $GL_n$. This is the structure lying behind formulas for branching multiplicities in terms of Littlewood-Richardson coefficients. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0407467v2-abstract-full').style.display = 'none'; document.getElementById('math/0407467v2-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 November, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 July, 2004; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2004. </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">39 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/math/0311159">arXiv:math/0311159</a> <span> [<a href="https://arxiv.org/pdf/math/0311159">pdf</a>, <a href="https://arxiv.org/ps/math/0311159">ps</a>, <a href="https://arxiv.org/format/math/0311159">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> </div> </div> <p class="title is-5 mathjax"> Stable branching rules for classical symmetric pairs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Howe%2C+R+E">Roger E. Howe</a>, <a href="/search/math?searchtype=author&query=Tan%2C+E+C">Eng Chye Tan</a>, <a href="/search/math?searchtype=author&query=Willenbring%2C+J+F">Jeb F. Willenbring</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="math/0311159v2-abstract-short" style="display: inline;"> We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of 10 classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewood's restriction rule… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0311159v2-abstract-full').style.display = 'inline'; document.getElementById('math/0311159v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="math/0311159v2-abstract-full" style="display: none;"> We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of 10 classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewood's restriction rule as a special case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('math/0311159v2-abstract-full').style.display = 'none'; document.getElementById('math/0311159v2-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 November, 2005; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 November, 2003; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2003. </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> 22E46 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601-1626 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

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