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The Waldspurger Transform of Permutations and Alternating Sign Matrices | | James McKeown
; | | Date: |
13 Jul 2017 | | 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). | | Source: | arXiv, 1707.3937 | | Services: | Forum | Review | PDF | Favorites | |
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