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Bubble sort and Howe duality for staircase matrices

Anton Khoroshkin, Ievgen Makedonskyi

TL;DR

The work tackles a longstanding problem in the interface of combinatorics and representation theory by giving an independent combinatorial proof of Cauchy-type identities for staircase matrices through a generalized bubble-sort on arborescent posets and a detailed study of DL-dense arrays. It develops a distributive-lattice framework for Demazure submodules, shows that Demazure submodules form a lattice with well-behaved subquotients (van der Kallen modules), and builds a comprehensive combinatorial model using staircase corners to realize a two-sided Howe duality for staircase matrices. The resulting identities express the character of symmetric algebras on staircase matrices as sums of Demazure-atom and key-polynomial terms, with a Möbius-function weighting on a DL-dense poset controlling the decomposition into minimal subquotients. The approach generalizes classical Pieri/Schur-type phenomena to the staircase setting and provides a coherent structure that intertwines Bruhat orders, EL-shellability, and Demazure module intersections, potentially informing broader Howe duality phenomena in non-rectangular geometries.

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.

Bubble sort and Howe duality for staircase matrices

TL;DR

The work tackles a longstanding problem in the interface of combinatorics and representation theory by giving an independent combinatorial proof of Cauchy-type identities for staircase matrices through a generalized bubble-sort on arborescent posets and a detailed study of DL-dense arrays. It develops a distributive-lattice framework for Demazure submodules, shows that Demazure submodules form a lattice with well-behaved subquotients (van der Kallen modules), and builds a comprehensive combinatorial model using staircase corners to realize a two-sided Howe duality for staircase matrices. The resulting identities express the character of symmetric algebras on staircase matrices as sums of Demazure-atom and key-polynomial terms, with a Möbius-function weighting on a DL-dense poset controlling the decomposition into minimal subquotients. The approach generalizes classical Pieri/Schur-type phenomena to the staircase setting and provides a coherent structure that intertwines Bruhat orders, EL-shellability, and Demazure module intersections, potentially informing broader Howe duality phenomena in non-rectangular geometries.

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.

Paper Structure

This paper contains 28 sections, 48 theorems, 146 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Recollection of (Bruhat) Posets and Distributive Lattices
  4. Recollection of poset terminology
  5. Bruhat Order and Bruhat Graph
  6. Distributive Lattice of Vector Subspaces
  7. Bubble-Sort Algorithm for Arborescent Posets with (Anti)Linearization
  8. Arborescent Posets with Consistent (Anti)Linearization
  9. Hasse diagram of $\mathsf{S}$-dominant compositions
  10. Bubble-sort
  11. The Desired Properties of the Posets ${\mathbb D}_{\mathsf{S}}(\lambda)$
  12. Case of Regular $\lambda$
  13. Parabolic case
  14. Arborescent Poset with Consistent Linearization
  15. Distributive lattice of Demazure submodules
  16. ...and 13 more sections

Key Result

Theorem D

For each partition $\lambda\vdash N$ with $l(\lambda)\leqslant \#\mathbf{S}{\mathbf{c}}_{\overline{n}}$, the following isomorphism of ${\mathfrak b}_n\text{-}{\mathfrak b}_m$-(bi)-submodules holds:

Theorems & Definitions (113)

  • Example B
  • Theorem D
  • Theorem F
  • Theorem H
  • Corollary I
  • Corollary J
  • Lemma 1.6
  • proof
  • Corollary 1.12
  • Corollary 1.14
  • ...and 103 more