Diagonalization of Matrices of Statistics

Abstract

In the work of Varchenko, Zagier, Thibon, Renteln, 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.