Diagonalization of the Matrices of the Multinomial Descent and Multinomial Inversion Statistics on the Symmetric Group

TL;DR

This paper investigates the spectral properties of multiplication maps on the symmetric group algebra associated with multinomial descent and inversion statistics, extending classical results and deriving explicit spectra and multiplicities.

Contribution

It introduces multinomial descent and inversion statistics, and determines the spectrum and multiplicities of the corresponding multiplication maps on the symmetric group algebra.

Findings

Spectra of multiplication maps for des_X and inv_X are explicitly determined.

Derived spectra for classical statistics des, maj, and inv as corollaries.

Provides new algebraic insights into permutation statistics via linear algebra.

Abstract

In the work of Varchenko, Zagier, Thibon, and Reiner, Saliola, Welker, linear algebraic properties of the multiplication map on the group algebra of the group algebra element are studied, which is the sum over all permutations weighted by q^{inv}, q^{maj}, inv. Here q is a variable, and inv and maj are the classical statistics inversion and major index. We define a multinomial descent statistic des_X and a multinomial inversion statistic inv_X. These new defined statistics are the multinomial expressions of the classical statistics descent des and inversion. We determine the spectrum and the multiplicity of each element of the spectrum of the analogously defined multiplication map on the group algebra for both des_X and inv_X. As corollaries we deduce the spectrum and the multiplicity of each element of the spectrum of the defined multiplication map on the group algebra for the statistics des, maj and inv.