On the relationship between combinatorial functions and representation theory

Abstract

The paper is devoted to the study of well-known combinatorial functions on the symmetric group S n —the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[S n ], and the restriction of the left regular representation of the group S n to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and 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 fix of fixed points.