Colored-descent representations of complex reflection groups G(r, p, n)

Abstract

We study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products Zr ≀ Sn, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.

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…”

“…whereas Theorem 10.5 in [6] asserts that the right-hand side above also describes the characters of Bagno and Biagioli's descent representations.…”

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

“…As applications we construct a basis for the coinvariant ring consisting of certain of the non-symmetric Jack polynomials discovered in [6], and give a new realization of the "colored descent representations" studied in [2] as irreducible modules for a generalized graded affine Hecke algebra.…”

“…As an application they decompose the coinvariant ring into "colored descent representations". Analagous results for the groups G(r, p, n) are contained in the paper [2] of Bagno and Biagoli. Our main theorem (5.2) constructs a new basis consisting of non-symmetric Jack polynomials by viewing the coinvariant ring as a module for the rational Cherednik algebra, and shows that upon restriction to the generalized graded affine Hecke algebra inside H, the coinvariant ring decomposes into irreducible submodules corresponding to "colored descent classes" of elements of G(r, p, n).…”

“…Our main theorem (5.2) constructs a new basis consisting of non-symmetric Jack polynomials by viewing the coinvariant ring as a module for the rational Cherednik algebra, and shows that upon restriction to the generalized graded affine Hecke algebra inside H, the coinvariant ring decomposes into irreducible submodules corresponding to "colored descent classes" of elements of G(r, p, n). These are the representations studied without the use of Hecke algebras in [2], and the leading terms of our basis elements are their "colored descent monomials". We suspect that a version of our results holds for Weyl groups, upon replacing the rational Cherednik algebra and Jack polynomials with the double affine Hecke algebra and Macdonald polynomials, and using the exponential coinvariant ring in place of the coinvariant ring.…”