Character formulas and descents for the hyperoctahedral group

Abstract

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.

The Eulerian Distribution on the Involutions of the Hyperoctahedral Group is Unimodal

“…One key ingredient of the proof of Theorem 1.1 is a B n -analogue of the well known expansion of the Schur symmetric functions in terms of fundamental quasisymmetric functions, obtained by Adin et al in [1] (discussed in Subsection 2.3).…”

“…This section fixes notation and briefly reviews background material regarding the combinatorics of (signed) permutations and Young (bi)tableaux, symmetric and unimodal polynomials and the theory of symmetric and quasisymmetric functions which will be needed in the sequel. More information on these topics can be found in [1], [14] and [15,Chapter 7].…”

“…Then, using Theorem 1.1, it derives a linear recurrence formula for the coefficients of I B n (x). Proof of (1). For a power series f (x), m ∈ Z >0 and indeterminate t we write f (t m ) = f (x 1 = x 2 = · · · = x m = t, x m+1 = x m+2 = · · · = 0).…”

Hultman Elements for the Hyperoctahedral Groups

“…Since conjugation by w 0 fixes w for any w ∈ B n , conjugating any cycle c in the cycle decomposition of w also gives a cycle c = w 0 cw 0 of w. If c = c then we let c = c = c and call c an odd cycle of w. If c = c, then we let c = cc and call c an even cycle of w. Alternatively, c is an odd cycle if there is an odd number of i such that i ≤ n and c(i) > n, and c is an even cycle if there is an even number of i such that i ≤ n and c(i) > n. It is easy to show that the reflection length of w (as an element of B n ) is n − ecyc(w), where ecyc(w) is the number of even cycles of w. (The cycle decomposition for elements of B n and its relation to reflection length have appeared in the literature many times. Our description of the cycle decomposition comes from [3] and our terminology from [22]. The fact about reflection length is stated in [9] and [17].…”

“…Let n = 3 and w = 426153. Then E(w) = {((3, 2),(5,2),(5,4),(3,4)}. We see that w is not defined by pseudo-inclusions since r w (3, 2) = 1 = max(0, 2 −3 + 1).…”