Beijing Institute of Mathematical Sciences and Applications

Shamil Shakirov

Yevhen Makedonskyi

Friday, October 25, 2024 1:00 PM - 2:30 PM

A3-4-301

Zoom 518 868 7656 (BIMSA)

The well-known Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $1 \leq i \leq n$ and $1 \leq j \leq m$ as a sum of products of Schur polynomials: $\sum_{\lambda\colon l(\lambda)\leq\min(m,n)} s_{\lambda}(x)s_{\lambda}(y)$. In algebraic terms, the identity represents the decomposition of the symmetric algebra of the space of rectangular matrices, viewed as a bi-module for the general linear Lie algebras $\gl_n -\gl_m$, where the algebras act by transforming rows and columns, respectively (Howe duality). We generalize the Cauchy kernel by breaking the symmetry and replacing rectangular matrices with arbitrary staircase-shaped matrices. Specifically, for any given staircase shape $S$, we present the formula that expand the product of terms $(1 - x_i y_j)^{-1}$, for $(i,j) \in S$, as a sum involving products of key polynomials and Demazure atoms. We give two representation-theoretic interpretations of these new identities involves different decompositions of the bi-module structure of the symmetric algebra of the space of staircase matrices, with the two Borel subalgebras acting by corresponding row and column transformations.

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.