Character formulas and descents for the hyperoctahedral group

A general setting to study a certain type of formulas, expressing characters of the symmetric group 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 Désarménien and Wachs.