Duality theorems for staircase matrices
演讲者
时间
2025年04月18日 13:00 至 14:30
地点
A3-4-301
线上
Zoom 242 742 6089
(BIMSA)
摘要
The well known Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $\lambda$ of products of Schur polynomials: $s_{\lambda}(x)s_{\lambda}(y)$. Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a $\mathfrak{gl}_m$-$\mathfrak{gl}_n$-bi-module.
I will talk about the generalization of such decompositions by replacing rectangular matrices with arbitrary staircase-shaped matrices equipped with the left and right actions of the Borel upper-triangular subalgebras.
For any given staircase shape $\mathsf{Y}$, we describe left and right "standard" filtrations on the symmetric algebra of the space of shape $\mathsf{Y}$ matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras.
On the level of characters, we derive three distinct expansions for the product $(1 - x_i y_j)^{-1}$ for $(i,j) \in \mathsf{Y}$. The first two expansions are sums of products of key polynomials $\kappa_\lambda(x)$ and (opposite) Demazure atoms $a^{\mu}(y)$. The third expansion is an alternating sum of products of key polynomials $\kappa_{\lambda}(x)\,\kappa^{\mu}(y)$.
The talk will be based on two papers, one of them is joint with Anton Khoroshkin and Evgeny Feigin and the second is joint with Anton Khoroshkin.
I will talk about the generalization of such decompositions by replacing rectangular matrices with arbitrary staircase-shaped matrices equipped with the left and right actions of the Borel upper-triangular subalgebras.
For any given staircase shape $\mathsf{Y}$, we describe left and right "standard" filtrations on the symmetric algebra of the space of shape $\mathsf{Y}$ matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras.
On the level of characters, we derive three distinct expansions for the product $(1 - x_i y_j)^{-1}$ for $(i,j) \in \mathsf{Y}$. The first two expansions are sums of products of key polynomials $\kappa_\lambda(x)$ and (opposite) Demazure atoms $a^{\mu}(y)$. The third expansion is an alternating sum of products of key polynomials $\kappa_{\lambda}(x)\,\kappa^{\mu}(y)$.
The talk will be based on two papers, one of them is joint with Anton Khoroshkin and Evgeny Feigin and the second is joint with Anton Khoroshkin.
演讲者介绍
Ievgen Makedonskyi于俄罗斯高等经济研究大学获得数学博士学位，先后在俄罗斯高等经济研究大学、马克斯普朗克数学研究所、东京大学、斯科尔科沃科技大学、德国耶拿大学任职，2022年加入北京雁栖湖应用数学研究院任助理研究员，研究兴趣包括李代数、多项式导子、仿射Kac-Moody李代数、Weyl和Demazure模、非对称Macdonald多项式、近世代数、弧簇等。