The waldspurger transform of permutations and alternating sign matrices

In 2005 J. L. Waldspurger proved the following theorem: Given a finite real reflection group G, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations 1 - g. Shortly after this E. Meinrenken extended the result to affine Weyl groups and then P. V. Bibikov and V. S. Zhgoon 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 an interesting new perspective on the combinatorics of the symmetric group. In particular, for each permutation matrix g G S_nwe define a non-negative integer matrix WT(g), called the Waldspurger transform of g. The definition of the matrix WT(g) is purely combinatorial but it turns out that its columns are the images of the fundamental weights under 1 - g, expressed in simple root coordinates. The possible columns of WT(g) (which we call UM vectors) biject to many interesting structures including: unimodal Motzkin paths, abelian ideals in the Lie algebra sln(C), Young diagrams with maximum hook length n, and integer points inside a certain polytope. We show that the sum of the entries of WT(g) is half the entropy of the corresponding permutation g, which is known to equal the rank of g in the MacNeille completion of the Bruhat order. Inspired by this, we extend the Waldspurger transform WT(M) to alternating sign matrices M and give an intrinsic characterization of the image. This provides a geometric realization of MacNeille completion of the Bruhat order (a.k.a. the lattice of alternating sign matrices).