Character formulas and descents for the hyperoctahedral group

TL;DR

This paper extends character formulas and descent set analysis from the symmetric group to the hyperoctahedral group, introducing new formulas, functions, and matrices with applications in algebraic combinatorics.

Contribution

It develops a new formula for irreducible characters of $B_n$, introduces signed quasisymmetric functions, and constructs Walsh--Hadamard type matrices for the hyperoctahedral group.

Findings

Formulas for $B_n$-actions on coinvariant and exterior algebras.

Characterizations of top homology of certain posets using signed permutations.

A $B_n$-analogue of an equidistribution theorem.

Abstract

A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is further investigated in this paper and extended to the hyperoctahedral group $B_n$. Key ingredients are a new formula for the irreducible characters of $B_n$, the signed quasisymmetric functions introduced by Poirier, and a new family of matrices of Walsh--Hadamard type. Applications include formulas for natural $B_n$-actions on coinvariant and exterior algebras and on the top homology of a certain poset in terms of the combinatorics of various classes of signed permutations, as well as a $B_n$-analogue of an equidistribution theorem of D\'esarm\'enien and Wachs.