北京雁栖湖应用数学研究院

沙米尔·沙基洛夫

叶夫根·马科东斯基

2024年10月25日 13:00 至 14:30

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。