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

Griffeth, Stephen

doi:10.1090/S0002-9939-08-09697-4

Jack polynomials and the coinvariant ring of $G(r,p,n)$
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Proc. Amer. Math. Soc. 137 (2009), 1621-1629 Request permission

Abstract:

We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\mathbb {H}$ and its generalized graded affine Hecke subalgebra $\mathcal {H}$. We construct a basis consisting of non-symmetric Jack polynomials and, using this basis, decompose the coinvariant ring into irreducible modules for $\mathcal {H}$. The basis consists of certain non-symmetric Jack polynomials whose leading terms are the “descent monomials” for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman as well as Bagno and Biagoli. The irreducible $\mathcal {H}$-submodules of the coinvariant ring are their “colored descent representations”.

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