Bubble sort and Howe duality for staircase matrices

TL;DR

This paper provides a new combinatorial proof of Cauchy identities for staircase matrices, explores their algebraic structure through Demazure modules, and generalizes Howe duality in this context.

Contribution

It introduces an independent combinatorial proof, generalizes parabolic Bruhat graphs, and extends Howe duality for staircase matrices.

Findings

New combinatorial proof of Cauchy identities

Insights into coefficients in the identities

Generalization of Howe duality for staircase matrices

Abstract

In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive new insights into certain coefficients appearing in one of these identities. The first part of the paper focuses on combinatorial aspects. It is self-contained, of independent interest, and introduces a generalization of parabolic Bruhat graphs for monotone functions on an arborescent poset. The second part examines the intersections of Demazure modules within a given integrable representation. Finally, we propose a generalization of the classical Howe duality for staircase matrices in terms of the corresponding distributive lattice of Demazure submodules. Computing the associated character yields the desired Cauchy identities for staircase matrices.