On the relation of some combinatorial functions to representation theory

TL;DR

This paper explores the connection between key combinatorial functions on the symmetric group and representation theory, revealing their shared algebraic structure and deriving simple matrix formulas for their analysis.

Contribution

It demonstrates that the major index, descent number, and inversion number generate the same ideal in the group algebra and links their representations to skew-symmetric matrices, providing new formulas and spectral insights.

Findings

All three functions generate the same ideal in the group algebra.

The restriction of the regular representation to this ideal is isomorphic to a skew-symmetric matrix representation.

Derived formulas for these functions in terms of simple matrix forms and identified their spectra.

Abstract

The paper is devoted to the study of some well-knonw combinatorial functions on the symmetric group $\sn$ --- the major index $\maj$, the descent number $\des$, and the inversion number $\inv$ --- from the representation-theoretic point of view. We show that each of these functions generates in the group algebra the same ideal, and the restriction of the left regular representation to this ideal is isomorphic to the representation of $\sn$ in the space of $n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the functions $\maj,\des,\inv$ in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as to deduce a series of identities relating these functions to one another and to the number of fixed points $\fix$. \smallskip\noindent {\it Keywords:} major index, descent number, inversion number, representations of the symmetric group, skew-symmetric matrices, dual complexity.