Bubble sort and Howe duality for staircase matrices

Abstract

We prove the alternating Cauchy identity for staircase matrices conjectured in arXiv:2411.03117, together with an explicit description of the coefficients occurring in it. As a byproduct, our approach also yields a new, independent (more combinatorial) proof of the Cauchy identities for staircase matrices established in arXiv:2411.03117. 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 centers on representation theory. We propose a generalization of the classical Howe duality for staircase matrices in terms of the distributive lattice of Demazure submodules within a given integrable representation. Computing the associated character yields all desired Cauchy identities for staircase matrices.