Descent representations and multivariate statistics

Abstract

Abstract. Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.

A Solomon descent theory for the wreath products $G\wr\mathfrak S_n$

“…for some Γ-composition c of n. This terminology agrees with that introduced by Adin, Brenti and Roichman in [1] for the case G = {±1} and extended by Bagno and Biagioli in [6] to the more general context of complex reflection groups G(r, p, n). Indeed formula (5.3) and the equalityθ G (t T ) = χ sh(T) yield the decomposition into irreducible characters…”

Descent representations for generalized coinvariant algebras

“…Descent representations µ = (3, 2, 2, 1), then the descents of µ are 1, 3, 4, and the multiplicity of S (2,2,1) is 1 since the only standard Young tableau of shape (2,2,1) with descent set {1, 3, 4} is 121 3 2 4…”

“…, x n with vanishing constant term. The set that is important for the generalization of R n that we are considering is the elementary symmetric functions (2) e d := Since I n is homogeneous and invariant under the action of S n , the coinvariant algebra is a graded S n -module. Since the conjugacy classes of S n are indexed by partitions of n, the irreducible representations of S n are also indexed by partitions of n (an explicit construction of irreducibles is given by Specht modules).…”

“…Adin, Brenti, and Roichman [2] show that R n,µ is zero unless µ is a partition with at most n − 1 parts such that the differences between consecutive parts are at most 1. We call such partitions descent partitions.…”

Jack polynomials and the coinvariant ring of 𝐺(𝑟,𝑝,𝑛)

“…In [1] Adin, Brenti and Roichman constructed "colored descent bases" and a corresponding "flag major index" for the coinvariant rings of the type B Weyl groups G(2, 1, n) (see [1]). As an application they decompose the coinvariant ring into "colored descent representations".…”