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* in **S**_{n} we 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
**sl**_{n}(**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).

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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