Duality theorems for staircase matrices
Organizers
Speaker
Time
Friday, April 18, 2025 1:00 PM - 2:30 PM
Venue
A3-4-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
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.
Speaker Intro
Ievgen Makedonskyi obtained a PhD degree in mathematics from the Russian University of Advanced Economic Research and then worked at the Russian University of Advanced Economic Research, the Max Planck Institute of Mathematics, the University of Tokyo, Skolkovo University of Science and Technology, and Jena University in Germany. In 2022, he joined the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications as an assistant professor. His research interests include Lie algebra, polynomial derivation, affine Kac Moody Lie algebra, Weyl and Demazure module Asymmetric Macdonald polynomials, recent algebras, arc varieties, etc.