The Waldspurger Transform of Permutations and Alternating Sign Matrices

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).