Cauchy identities for staircase matrices

TL;DR

This paper generalizes the classical Cauchy identity from rectangular matrices to staircase-shaped matrices, providing new filtrations and character expansions involving Demazure modules and key polynomials.

Contribution

It introduces a new decomposition for staircase matrices, extending the Cauchy identity with filtrations and character formulas involving Demazure and van der Kallen modules.

Findings

Derived three new expansions for the product (1 - x_i y_j)^{-1} over staircase shapes.

Established filtrations on symmetric algebras of staircase matrices with Demazure module subquotients.

Connected character formulas to key polynomials and Demazure atoms in the context of staircase matrices.

Abstract

The celebrated 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. We generalize the Cauchy decomposition 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 $\kappa_{\lambda}(x)\,\kappa^{\mu}(y)$.