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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/2502.10912">arXiv:2502.10912</a> <span> [<a href="https://arxiv.org/pdf/2502.10912">pdf</a>, <a href="https://arxiv.org/ps/2502.10912">ps</a>, <a href="https://arxiv.org/format/2502.10912">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"> Orbits on a product of two flags and a line and the Bruhat Order, I </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=Evens%2C+S">Sam Evens</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="2502.10912v1-abstract-short" style="display: inline;"> Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\mathcal{B}_{n}$ be its flag variety. The standard Borel subgroup $B$ of upper triangular matrices acts on the product $\mathcal{B}_{n}\times \mathbb{P}^{n-1}$ with finitely many orbits. In this paper, we study the $B$-orbits on the subvarieties $\mathcal{B}_{n}\times \mathcal{O}_{i}$, where $\mathcal{O}_{i}$ is the $B$-orbit o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.10912v1-abstract-full').style.display = 'inline'; document.getElementById('2502.10912v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.10912v1-abstract-full" style="display: none;"> Let $G=GL(n)$ be the $n\times n$ complex general linear group and let $\mathcal{B}_{n}$ be its flag variety. The standard Borel subgroup $B$ of upper triangular matrices acts on the product $\mathcal{B}_{n}\times \mathbb{P}^{n-1}$ with finitely many orbits. In this paper, we study the $B$-orbits on the subvarieties $\mathcal{B}_{n}\times \mathcal{O}_{i}$, where $\mathcal{O}_{i}$ is the $B$-orbit on $\mathbb{P}^{n-1}$ containing the line through the origin in the direction of the $i$-th standard basis vector of $\mathbb{C}^{n}$. For each $i=1,\dots, n$, we construct a bijection between $B$-orbits on $\mathcal{B}_{n}\times\mathcal{O}_{i}$ and certain pairs of Schubert cells in $\mathcal{B}_{n}\times\mathcal{B}_{n}$. We also show that this bijection can be used to understand the Richardson-Springer monoid action on such $B$-orbits in terms of the classical monoid action of the symmetric group on itself. We also develop combinatorial models of these orbits and use these models to compute exponential generating functions for the sequences $\{|B\backslash(\mathcal{B}_{n}\times\mathcal{O}_{i})|\}_{n\geq 1}$ and $\{|B\backslash (\mathcal{B}_{n}\times \mathbb{P}^{n-1})|\}_{n\geq 1}$. In the sequel to this paper, we use the results of this paper to construct a correspondence between $B$-orbits on $\mathcal{B}_{n}\times\mathbb{P}^{n-1}$ and a collection of $B$-orbits on the flag variety $\mathcal{B}_{n+1}$ of $GL(n+1)$ and show that this correspondence respects closures relations and preserves monoid actions. As a consequence both closure relations and monoid actions for all $B$-orbits on $\mathcal{B}_{n}\times\mathbb{P}^{n-1}$ can be understood via the Bruhat order by using our results in [CE]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.10912v1-abstract-full').style.display = 'none'; document.getElementById('2502.10912v1-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 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">24 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15; 14L30; 20G20; 05E14 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.10122">arXiv:2406.10122</a> <span> [<a href="https://arxiv.org/pdf/2406.10122">pdf</a>, <a href="https://arxiv.org/ps/2406.10122">ps</a>, <a href="https://arxiv.org/format/2406.10122">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"> $B_{n-1}$-orbits on the flag variety and the Bruhat graph for $S_{n}\times S_{n}$ </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=Evens%2C+S">Sam Evens</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="2406.10122v1-abstract-short" style="display: inline;"> Let $G=G_{n}=GL(n)$ be the $n\times n$ complex general linear group and embed $G_{n-1}=GL(n-1)$ in the top left hand corner of $G$. The standard Borel subgroup of upper triangular matrices $B_{n-1}$ of $G_{n-1}$ acts on the flag variety of $G$ with finitely many orbits. In this paper, we show that each $B_{n-1}$-orbit is the intersection of orbits of two Borel subgroups of $G$ acting on the flag v… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10122v1-abstract-full').style.display = 'inline'; document.getElementById('2406.10122v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.10122v1-abstract-full" style="display: none;"> Let $G=G_{n}=GL(n)$ be the $n\times n$ complex general linear group and embed $G_{n-1}=GL(n-1)$ in the top left hand corner of $G$. The standard Borel subgroup of upper triangular matrices $B_{n-1}$ of $G_{n-1}$ acts on the flag variety of $G$ with finitely many orbits. In this paper, we show that each $B_{n-1}$-orbit is the intersection of orbits of two Borel subgroups of $G$ acting on the flag variety of $G$. This allows us to give a new combinatorial description of the $B_{n-1}$-orbits by associating to each orbit a pair of Weyl group elements. The closure relations for the $B_{n-1}$-orbits can then be understood in terms of the Bruhat order on the Weyl group, and the Richardson-Springer monoid action on the orbits can be understood in terms of the classical monoid action of the Weyl group on itself. This approach makes the closure relation more transparent than in earlier work of Magyar and the monoid action significantly more computable than in our earlier papers, and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10122v1-abstract-full').style.display = 'none'; document.getElementById('2406.10122v1-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 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">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15; 14L30; 20G20; 05E14 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2107.10819">arXiv:2107.10819</a> <span> [<a href="https://arxiv.org/pdf/2107.10819">pdf</a>, <a href="https://arxiv.org/ps/2107.10819">ps</a>, <a href="https://arxiv.org/format/2107.10819">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"> B_{n-1}-bundles on the flag variety, II </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=Evens%2C+S">Sam Evens</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="2107.10819v1-abstract-short" style="display: inline;"> This paper is the sequel to ``$B_{n-1}$-bundles on the flag variety, I". We continue our study of the orbits of a Borel subgroup $B_{n-1}$ of $G_{n-1}=GL(n-1)$ (resp. $SO(n-1)$) acting on the flag variety $\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). We begin by using the results of the first paper to obtain a complete combinatorial model of the $B_{n-1}$-orbits on $\mathcal{B}_{n}$ in terms of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.10819v1-abstract-full').style.display = 'inline'; document.getElementById('2107.10819v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.10819v1-abstract-full" style="display: none;"> This paper is the sequel to ``$B_{n-1}$-bundles on the flag variety, I". We continue our study of the orbits of a Borel subgroup $B_{n-1}$ of $G_{n-1}=GL(n-1)$ (resp. $SO(n-1)$) acting on the flag variety $\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). We begin by using the results of the first paper to obtain a complete combinatorial model of the $B_{n-1}$-orbits on $\mathcal{B}_{n}$ in terms of partitions into lists. The model allows us to obtain explicit formulas for the number of orbits as well as the exponential generating functions for the sequences $\{|B_{n-1}\backslash \mathcal{B}_{n}|\}_{n\geq 1}$ . We then use the combinatorial description of the orbits to construct a canonical set of representatives of the orbits in terms of flags. These representatives allow us to understand an extended monoid action on $B_{n-1}\backslash \mathcal{B}_{n}$ using simple roots of both $\mathfrak{g}_{n-1}$ and $\mathfrak{g}$ and show that the closure ordering on $B_{n-1}\backslash \mathcal{B}_{n}$ is the standard ordering of Richardson and Springer. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.10819v1-abstract-full').style.display = 'none'; document.getElementById('2107.10819v1-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 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">45 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15; 14L30; 20G20; 05A15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.02405">arXiv:2102.02405</a> <span> [<a href="https://arxiv.org/pdf/2102.02405">pdf</a>, <a href="https://arxiv.org/ps/2102.02405">ps</a>, <a href="https://arxiv.org/format/2102.02405">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"> B_{n-1}-bundles on the flag variety, I </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=Evens%2C+S">Sam Evens</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="2102.02405v3-abstract-short" style="display: inline;"> We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.02405v3-abstract-full').style.display = 'inline'; document.getElementById('2102.02405v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.02405v3-abstract-full" style="display: none;"> We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify $B_{n-1}$-orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and determine completely the closure relation between orbit closures. This further develops work of Hashimoto in the general linear group case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.02405v3-abstract-full').style.display = 'none'; document.getElementById('2102.02405v3-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 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">24 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15 (Primary) </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/1808.04424">arXiv:1808.04424</a> <span> [<a href="https://arxiv.org/pdf/1808.04424">pdf</a>, <a href="https://arxiv.org/ps/1808.04424">ps</a>, <a href="https://arxiv.org/format/1808.04424">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"> The Complex Orthogonal Gelfand-Zeitlin System </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=Evens%2C+S">Sam Evens</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="1808.04424v1-abstract-short" style="display: inline;"> In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand-Zeitlin system. We show that the moment map (orthogonal Kostant-Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand-Zeitlin system on regular adjoint… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.04424v1-abstract-full').style.display = 'inline'; document.getElementById('1808.04424v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.04424v1-abstract-full" style="display: none;"> In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand-Zeitlin system. We show that the moment map (orthogonal Kostant-Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand-Zeitlin system on regular adjoint orbits and describe the generic flows of the integrable system. We also study the nilfibre of the moment map and show that in contrast to the general linear case it contains no strongly regular elements. This extends results of Kostant, Wallach, and Colarusso from the general linear case to the orthogonal case. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.04424v1-abstract-full').style.display = 'none'; document.getElementById('1808.04424v1-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, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">34 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L35; 14L30; 37J35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1410.3901">arXiv:1410.3901</a> <span> [<a href="https://arxiv.org/pdf/1410.3901">pdf</a>, <a href="https://arxiv.org/ps/1410.3901">ps</a>, <a href="https://arxiv.org/format/1410.3901">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"> Eigenvalue Coincidences and Multiplicity Free Spherical Pairs </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=Evens%2C+S">Sam Evens</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="1410.3901v2-abstract-short" style="display: inline;"> In recent work, we related the structure of subvarieties of $n\times n$ complex matrices defined by eigenvalue coincidences to $GL(n-1,\mathbb{C})$-orbits on the flag variety of $\mathfrak{gl}(n,\mathbb{C})$. In the first part of this paper, we extend these results to the complex orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n,\mathbb{C})$. In the second part of the paper, we use these result… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.3901v2-abstract-full').style.display = 'inline'; document.getElementById('1410.3901v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1410.3901v2-abstract-full" style="display: none;"> In recent work, we related the structure of subvarieties of $n\times n$ complex matrices defined by eigenvalue coincidences to $GL(n-1,\mathbb{C})$-orbits on the flag variety of $\mathfrak{gl}(n,\mathbb{C})$. In the first part of this paper, we extend these results to the complex orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n,\mathbb{C})$. In the second part of the paper, we use these results to study the geometry and invariant theory of the $K$-action on $\mathfrak{g}$, in the cases where $(\mathfrak{g}, K)$ is $(\mathfrak{gl}(n,\mathbb{C}), GL(n-1,\mathbb{C}))$ or $(\mathfrak{so}(n,\mathbb{C}), SO(n-1,\mathbb{C}))$. We study the geometric quotient $\mathfrak{g}\to \mathfrak{g}//K$ and describe the closed $K$-orbits on $\mathfrak{g}$ and the structure of the zero fibre. We also prove that for $x\in \mathfrak{g}$, the $K$-orbit $Ad(K)\cdot x$ has maximal dimension if and only if the algebraically independent generators of the invariant ring $\mathbb{C}[\mathfrak{g}]^{K}$ are linearly independent at $x$, which extends a theorem of Kostant. We give applications of our results to the Gelfand-Zeitlin system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.3901v2-abstract-full').style.display = 'none'; document.getElementById('1410.3901v2-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 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2014. </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">38 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15; 14L30; 20G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1309.5653">arXiv:1309.5653</a> <span> [<a href="https://arxiv.org/pdf/1309.5653">pdf</a>, <a href="https://arxiv.org/ps/1309.5653">ps</a>, <a href="https://arxiv.org/format/1309.5653">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"> Lie-Poisson theory for direct limit Lie algebras </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=Lau%2C+M">Michael Lau</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="1309.5653v1-abstract-short" style="display: inline;"> In this paper, we develop the fundamentals of Lie-Poisson theory for direct limits $G=\dirlim G_{n}$ of complex algebraic groups $G_{n}$ and their Lie algebras $\fg=\dirlim \fg_{n}$. We show that $\fg^{*}=\invlim\fg_{n}^{*}$ has the structure of a Poisson provariety and that each coadjoint orbit of $G$ on $\fg^{*}$ has the structure of an ind-variety. We construct a weak symplectic form on every c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.5653v1-abstract-full').style.display = 'inline'; document.getElementById('1309.5653v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1309.5653v1-abstract-full" style="display: none;"> In this paper, we develop the fundamentals of Lie-Poisson theory for direct limits $G=\dirlim G_{n}$ of complex algebraic groups $G_{n}$ and their Lie algebras $\fg=\dirlim \fg_{n}$. We show that $\fg^{*}=\invlim\fg_{n}^{*}$ has the structure of a Poisson provariety and that each coadjoint orbit of $G$ on $\fg^{*}$ has the structure of an ind-variety. We construct a weak symplectic form on every coadjoint orbit and prove that the coadjoint orbits form a weak symplectic foliation of the Poisson provariety $\fg^{*}$. We apply our results to the specific setting of $G=GL(\infty)=\dirlim GL(n,\C)$ and $\fg^{*}= M(\infty)=\invlim \fgl(n,\C)$, the space of infinite complex matrices with arbitrary entries. We construct a Gelfand-Zeitlin integrable system on $M(\infty)$, which generalizes the one constructed by Kostant and Wallach on $\fgl(n,\C)$. The system integrates to an action of a direct limit group $A(\infty)$ on $M(\infty)$, whose generic orbits are Lagrangian ind-subvarieties of the corresponding coadjoint orbit of $GL(\infty)$ on $M(\infty)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.5653v1-abstract-full').style.display = 'none'; document.getElementById('1309.5653v1-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 September, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2013. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L30; 20G20; 37J35; 53D17; 17B65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1303.6661">arXiv:1303.6661</a> <span> [<a href="https://arxiv.org/pdf/1303.6661">pdf</a>, <a href="https://arxiv.org/ps/1303.6661">ps</a>, <a href="https://arxiv.org/format/1303.6661">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</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"> Eigenvalue Coincidences and $K$-orbits, I </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=Evens%2C+S">Sam Evens</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="1303.6661v2-abstract-short" style="display: inline;"> We study the variety $\mathfrak{g}(l)$ consisting of matrices $x \in \mathfrak{gl}(n,\C)$ such that $x$ and its $n-1$ by $n-1$ cutoff $x_{n-1}$ share exactly $l$ eigenvalues, counted with multiplicity. We determine the irreducible components of $\mathfrak{g}(l)$ by using the orbits of $GL(n-1,\C)$ on the flag variety $\B_n$ of $\mathfrak{gl}(n,\C)$. More precisely, let $\mathfrak{b} \in \B_n$ be a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.6661v2-abstract-full').style.display = 'inline'; document.getElementById('1303.6661v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1303.6661v2-abstract-full" style="display: none;"> We study the variety $\mathfrak{g}(l)$ consisting of matrices $x \in \mathfrak{gl}(n,\C)$ such that $x$ and its $n-1$ by $n-1$ cutoff $x_{n-1}$ share exactly $l$ eigenvalues, counted with multiplicity. We determine the irreducible components of $\mathfrak{g}(l)$ by using the orbits of $GL(n-1,\C)$ on the flag variety $\B_n$ of $\mathfrak{gl}(n,\C)$. More precisely, let $\mathfrak{b} \in \B_n$ be a Borel subalgebra such that the orbit $GL(n-1,\C)\cdot \mathfrak{b}$ in $\B_n$ has codimension $l$. Then we show that the set $Y_{\fb}:= \{\Ad(g)(x): x\in \mathfrak{b} \cap \mathfrak{g}(l), g\in GL(n-1,\C)\}$ is an irreducible component of $\mathfrak{g}(l)$, and every irreducible component of of $\mathfrak{g}(l)$ is of the form $Y_{\mathfrak{b}}$, where $\mathfrak{b}$ lies in a $GL(n-1,\C)$-orbit of codimension $l$. An important ingredient in our proof is the flatness of a variant of a morphism considered by Kostant and Wallach, and we prove this flatness assertion using ideas from symplectic geometry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.6661v2-abstract-full').style.display = 'none'; document.getElementById('1303.6661v2-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 April, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2013. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M15; 14L30; 20G20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1111.2868">arXiv:1111.2868</a> <span> [<a href="https://arxiv.org/pdf/1111.2868">pdf</a>, <a href="https://arxiv.org/ps/1111.2868">ps</a>, <a href="https://arxiv.org/format/1111.2868">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"> The Gelfand-Zeitlin integrable system and K-orbits on the flag variety </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=Evens%2C+S">Sam Evens</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="1111.2868v1-abstract-short" style="display: inline;"> In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of $n\times n$ complex matrices $\fgl(n,\C)$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.2868v1-abstract-full').style.display = 'inline'; document.getElementById('1111.2868v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1111.2868v1-abstract-full" style="display: none;"> In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of $n\times n$ complex matrices $\fgl(n,\C)$ introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of $K_{n}=GL(n-1,\C)\times GL(1,\C)$-orbits on the flag variety $\mathcal{B}_{n}$ of $GL(n,\C)$ to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of $K_{n}$ and $GL(n,\C)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1111.2868v1-abstract-full').style.display = 'none'; document.getElementById('1111.2868v1-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 November, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2011. </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">33 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/1105.1726">arXiv:1105.1726</a> <span> [<a href="https://arxiv.org/pdf/1105.1726">pdf</a>, <a href="https://arxiv.org/ps/1105.1726">ps</a>, <a href="https://arxiv.org/format/1105.1726">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="Symplectic Geometry">math.SG</span> </div> </div> <p class="title is-5 mathjax"> K-orbits on the flag variety and strongly regular nilpotent matrices </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=Evens%2C+S">Sam Evens</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="1105.1726v1-abstract-short" style="display: inline;"> In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.1726v1-abstract-full').style.display = 'inline'; document.getElementById('1105.1726v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1105.1726v1-abstract-full" style="display: none;"> In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i-1,\C) \times GL(1,\C)$-orbits on the flag variety for $\fgl(i,\C)$ for $2\leq i\leq n+1$. As a consequence, we obtain a more precise description of the nilfibre. The $K_{i}$-orbits contributing to the nilfibre are closely related to holomorphic and anti-holomorphic discrete series for the real Lie groups $U(i,1)$, with $i \le n$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.1726v1-abstract-full').style.display = 'none'; document.getElementById('1105.1726v1-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, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2011. </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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20G20; 53D17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0908.3879">arXiv:0908.3879</a> <span> [<a href="https://arxiv.org/pdf/0908.3879">pdf</a>, <a href="https://arxiv.org/ps/0908.3879">ps</a>, <a href="https://arxiv.org/format/0908.3879">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Symplectic Geometry">math.SG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> On Algebraic Integrability of Gelfand-Zeitlin fields </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=Evens%2C+S">Sam Evens</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="0908.3879v1-abstract-short" style="display: inline;"> We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to stratify the strongly regular set by subvarieties $X_{D}$. We construct an 茅tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and $\hat{\mathfrak{g}}$ are smooth… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0908.3879v1-abstract-full').style.display = 'inline'; document.getElementById('0908.3879v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0908.3879v1-abstract-full" style="display: none;"> We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in $gl(n,\mathbb{C})$. We use decomposition classes to stratify the strongly regular set by subvarieties $X_{D}$. We construct an 茅tale cover $\hat{\mathfrak{g}}$ of $X_{D}$ and show that $X_{D}$ and $\hat{\mathfrak{g}}$ are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on $X_{D}$ to Hamiltonian vector fields on $\hat{\mathfrak{g}}$ and integrate these vector fields to an action of a connected, commutative algebraic group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0908.3879v1-abstract-full').style.display = 'none'; document.getElementById('0908.3879v1-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, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 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">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L30; 53D17; 20G20; 37J35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0811.1351">arXiv:0811.1351</a> <span> [<a href="https://arxiv.org/pdf/0811.1351">pdf</a>, <a href="https://arxiv.org/ps/0811.1351">ps</a>, <a href="https://arxiv.org/format/0811.1351">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Symplectic Geometry">math.SG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The orbit structure of the Gelfand-Zeitlin group on n x n matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Colarusso%2C+M">Mark Colarusso</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="0811.1351v2-abstract-short" style="display: inline;"> In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq \mathbb{C}^{n\choose 2}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the authors show that $A$-orbits of dimension ${n\choose 2}$ form Lagrangian submanifolds of r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.1351v2-abstract-full').style.display = 'inline'; document.getElementById('0811.1351v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0811.1351v2-abstract-full" style="display: none;"> In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq \mathbb{C}^{n\choose 2}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the authors show that $A$-orbits of dimension ${n\choose 2}$ form Lagrangian submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit structure of $A$ on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all $A$-orbits of dimension ${n\choose 2}$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.1351v2-abstract-full').style.display = 'none'; document.getElementById('0811.1351v2-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 March, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 9 November, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2008. </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">30 pages: Version 2 contains a stronger result in section 5.3 (Theorem 5.15)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L30; 14R20; 37J35; 53D17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0811.0835">arXiv:0811.0835</a> <span> [<a href="https://arxiv.org/pdf/0811.0835">pdf</a>, <a href="https://arxiv.org/ps/0811.0835">ps</a>, <a href="https://arxiv.org/format/0811.0835">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Symplectic Geometry">math.SG</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"> The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Colarusso%2C+M">Mark Colarusso</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="0811.0835v1-abstract-short" style="display: inline;"> In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group $A$ isomorphic to \mathbb{C}^{n\choose 2} on gl(n) using a completely integrable system derived from Gelfand-Zeitlin theory. In this paper we show that an analogous action of \mathbb{C}^{d} exists on the complex orthogonal Lie algebra so(n), where d is half the dimen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.0835v1-abstract-full').style.display = 'inline'; document.getElementById('0811.0835v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0811.0835v1-abstract-full" style="display: none;"> In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group $A$ isomorphic to \mathbb{C}^{n\choose 2} on gl(n) using a completely integrable system derived from Gelfand-Zeitlin theory. In this paper we show that an analogous action of \mathbb{C}^{d} exists on the complex orthogonal Lie algebra so(n), where d is half the dimension of a regular adjoint orbit in so(n). In [KW1], Kostant and Wallach describe the orbits of $A$ on a certain Zariski open subset of regular semisimple elements in gl(n). We extend these results to the case of so(n). We also make brief mention of the author's results in [Col1], which describe all $A$-orbits of dimension {n\choose 2} in gl(n). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0811.0835v1-abstract-full').style.display = 'none'; document.getElementById('0811.0835v1-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, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2008. </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">27 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L30; 14R20; 37K10; 53D17 </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 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