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 Sn 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 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).
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