The Waldspurger Transform of Permutations and Alternating Sign Matrices

TL;DR

This paper introduces the Waldspurger transform for permutations and alternating sign matrices, revealing deep combinatorial connections and providing a geometric realization of the Bruhat order's Dedekind-MacNeille completion.

Contribution

It defines a combinatorial Waldspurger transform for permutations and extends it to alternating sign matrices, linking to various combinatorial structures and the Bruhat order.

Findings

The sum of entries in the transform equals permutation entropy.

Transform columns correspond to Motzkin paths, Young diagrams, and polytope points.

Provides a geometric interpretation of the Bruhat order's lattice.

Abstract

In 2005 J.L. Waldspurger proved the following theorem: given a finite real reflection group $W$, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations $id-w$. Shortly thereafter E. Meinrencken extended the result to affine Weyl groups. P.V. Bibikov and V.S. Zhgoon then gave a uniform proof for a discrete reflection group acting on a simply-connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give a new perspective on the combinatorics of the symmetric group. In particular, for each permutation matrix $w \in \mathfrak{S}_n$ we define a non-negative integer matrix $\mathbf{WT}(w)$, called the Waldspurger transform of $w$. The definition of the matrix $\mathbf{WT}(w)$ is purely combinatorial but its columns are the images of the fundamental weights under the action of $id-w$, expressed in simple root coordinates. The possible columns of $\mathbf{WT}(w)$ (which we call UM vectors) are in bijection with many interesting structures including: unimodal Motzkin paths, abelian ideals in nilradical of the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$, Young diagrams with maximum hook length $n$, and integer points inside a certain polytope. We show that the sum of the entries of $\mathbf{WT}(w)$ is equal to half the entropy of the corresponding permutation $w$, which is known to equal the rank of $w$ in the Dedekind-MacNeille completion of the Bruhat order. Inspired by this, we extend the Waldpurger transform $\mathbf{WT}(M)$ to alternating sign matrices $M$ and give an intrinsic characterization of the image. This provides a geometric realization of Dedekind-MacNeille completion of the Bruhat order (a.k.a. the lattice of alternating sign matrices).