首页/论文/math/0008109

arXiv math/00081092000-08-15math.RTmath.QA

关于Schur-Howe-Sergeev对偶的注记

Remarks on the Schur-Howe-Sergeev Duality

Shun-Jen Cheng, Weiqiang Wang

备注：: 11 pages, to appear in Lett. Math. Phys
发表信息：: Lett. Math. Phys. 52 (2000)143--153

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

摘要：

在两个奇异李超代数（q（m），q（n））之间建立了一个新的Howe对偶.这给出了一个著名的舒尔Q-函数的组合恒等式的表示论解释。我们进一步建立了这个新的Howe对偶与q（n）与超八面体群H_k的中心扩张$\Hy_k$之间的Schur-Sergeev对偶的等价性.证明了由n的严格划分给出的具有最高权$λ$的q（n）-模的零权空间是由$λ$参数化的有限群$\Hy_n$上的不可约模.我们还讨论了这种豪对偶的一些后果。

英文摘要：

We establish a new Howe duality between a pair of two queer Lie superalgebras (q(m),q(n)). This gives a representation theoretic interpretation of a well-known combinatorial identity for Schur Q-functions. We further establish the equivalence between this new Howe duality and the Schur-Sergeev duality between q(n) and a central extension $\Hy_k$ of the hyperoctahedral group H_k. We show that the zero-weight space of a q(n)-module with highest weight $λ$ given by a strict partition of n is an irreducible module over the finite group $\Hy_n$ parameterized by $λ$. We also discuss some consequences of this Howe duality.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The integral Schur-Weyl-Sergeev duality

Degenerating the quantum queer Schur superalgebra ${\mathcal{Q}_q(n,r; R)}$ to the case $q=1$, the queer Schur superalgebra ${\mathcal{Q}(n,r)}$ is obtained. In this article, we reconstruct the universal enveloping algebra ${U({\mathfrak{q}_n})}$ of the queer Lie superalgebra ${\mathfrak{q}_n}$ via ${\mathcal{Q}(n,r)}$, and achieve another explanation of the Schur-Weyl-Sergeev duality. Finally, we depict the Schur-Weyl-Sergeev duality over $\mathbb{Z}$.

2021-06-03math.QAmath.RT

Schur-Sergeev duality for Ariki-Koike algebras

Let $U_q(\mathfrak{g})$ be the quantized superalgebra of $\mathfrak{g}=\mathfrak{gl}(k_1|\ell_1)\oplus\cdots\oplus\mathfrak{gl}(k_m|\ell_m)$ and $H_{m,n}(q,\mathbf{Q})$ the cyclotomic Hecke algebra of type $G(m,1,n)$. We define a right $H_{m,n}(q,\mathbf{Q})$-action on the $n$-fold tensor (super)space of the vector representation of $U_q(\mathfrak{g})$ and prove the Schur--Weyl reciprocity between $U_q(\mathfak{g})$ and $H_{m,n}(q,\mathbf{Q})$.

2022-05-23math.QAmath.RT

Levi-type Schur-Sergeev duality for general linear super groups

In this note, we investigate a kind of double centralizer property for general linear supergroups. For the super space $V=\mathbb{K}^{m\mid n}$ over an algebraically closed field $\mathbb{K}$ whose characteristic is not equal to $2$, we consider its $\mathbb{Z}_2$-homogeneous one-dimensional extension $\underline V=V\oplus\mathbb{K}v$, and the natural action of the supergroup $\tilde G:=\text{GL}(V)\times \textbf{G}_m$ on $\underline V$. Then we have the tensor product supermodule ($\underline{V}^{\otimes r}$, $ρ_r$) of $\tilde G$. We present a kind of generalized Schur-Sergeev duality which i…

2022-06-30math.RT

主有限$W$-超代数的超Vust定理与Schur-Sergeev对偶

Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras

该研究针对一般线性李超代数提出超Vust定理，并以此为工具得到主有限$W$-超代数的Schur-Sergeev对偶，部分推广了Brundan-Kleshchev高阶Schur-Weyl对偶。

2025-03-23math.RT

Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mat…

2012-08-25math.COmath.RT

A–D型经典群特征标的第九变体与Littlewood恒等式

Ninth variation of classical group characters of type A-D and Littlewood identities

本文推广经典李群特征标与有理Schur函数，用任意多项式序列替代阶乘幂，推导Littlewood型恒等式并获得新的Jacobi–Trudi与Nägelsbach–Kostka恒等式。

2023-12-05math.COmath.RT

更多相关论文收起更多论文

Schur代数的几何构造

Geometric construction of Schur algebras

本文通过广义Steinberg簇的Borel-Moore同调与等变K群构造了任意型的广义Schur代数，并以此建立了局部几何Langlands对应的类似物、仿射AIII型拟分裂$\imath$量子群的实现及几何Howe对偶。

2024-12-22math.QAmath.RT

量子Howe对偶与类型AIII的Schur对偶

Quantum Howe duality and Schur duality of type AIII

The University of Hong Kong（香港大学）；New Cornerstone Science Laboratory（新基石科学实验室）

本文建立了一种新的联系，将Luo-Xu提出的类型AIII的iHowe对偶与Bao-Wang提出的iSchur对偶联系起来，证明了i权重空间与张量空间的自然同构，并展示了相对辫群作用与类型B Hecke代数作用的一致性。

2026-01-10math.QAmath.RT

Howe Duality for Quantum Queer Superalgebras

We establish a new Howe duality between a pair of quantum queer superalgebras $(\mathrm{U}_{q^{-1}}(\mathfrak{q}_n), \mathrm{U}_q(\mathfrak{q}_m))$. The key ingredient is the construction of a non-commutative analogue $\mathcal{A}_q(\mathfrak{q}_n,\mathfrak{q}_m)$ of the symmetric superalgebra $S(\mathbb{C}^{mn|mn})$ with the use of quantum coordinate queer superalgebra. It turns out that this superalgebra is equipped with a $\mathrm{U}_{q^{-1}}(\mathfrak{q}_n)\otimes\mathrm{U}_q(\mathfrak{q}_m)$-supermodule structure that admits a multiplicity-free decomposition. We also show that the $(\math…

2018-05-13math.RT

Peter-Weyl, Howe and Schur-Weyl theorems for current groups

The classical Peter-Weyl theorem describes the structure of the space of functions on a semi-simple algebraic group. On the level of characters (in type A) this boils down to the Cauchy identity for the products of Schur polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for the current groups. In particular, in type A the corresponding characters identity is governed by the Cauchy identity for the products of q-Whittaker functions. We also formulate and prove a version of the Schur-Weyl theorem for current groups. The link between the Peter-Weyl and Schur-Weyl theorems…

2019-06-07math.AGmath.COmath.RT

Geometric Howe dualities of finite type

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp. type B/C), the Howe duality between a pair of quantum general linear groups (resp. a pair of $\imath$quantum groups of type AIII/IV) is established. The Howe duality for quantum general linear groups has been provided via quantum coordinate algebras in [Z02]. We also generalize this algebraic approach to $\imath$quantum groups of type AIII/IV, and prove that…

2022-10-11math.QAmath.RT

Limit shapes for skew Howe duality

We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge asymptotic results for the first parts in the form of the Tracy--Widom distribution, as well as less-universal critical regime results introduced by Gravner, Tracy and Widom. We do this for a large class of Schur parameters going beyond the Plancherel or principal specializations previously st…

2022-11-24math-phmath.COmath.MP

$(GL_n, GL_k)$ 斜霍韦对偶性的极限形状与涨落

Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality

Université d’Angers（昂热大学）；St. Petersburg State University（圣彼得堡国立大学）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京应用数学与交叉科学研究院）；Hokkaido University（北海道大学）

本文研究双柯西恒等式中Schur多项式特化诱导的Young图概率测度，证明极限形状存在，并刻画体相与边界涨落分别由离散正弦核和Tracy-Widom分布等控制，恢复已知离散分布并发现Pearcey核出现条件。

2024-08-21math-phmath.COmath.MP