On the Relationship between Combinatorial Functions and Representation Theory

The paper is devoted to the study of well-known combinatorial functions on the symmetric group $\mathfrak{S}_n$—the major index $\operatorname{maj}$, the descent number $\operatorname{des}$, and the inversion number $\operatorname{inv}$—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra $\mathbb{C}[\mathfrak{S}_n]$, and the restriction of the left regular representation of the group $\mathfrak{S}_n$ to this ideal is isomorphic to its representation in the space of $n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the functions $\operatorname{maj}$, $\operatorname{des}$, and $\operatorname{inv}$ 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 derive a series of identities relating these functions to one another and to the number $\operatorname{fix}$ of fixed points.