Descent Representations of Generalized Coinvariant Algebras

Abstract

The coinvariant algebra Rn is a well-studied Sn-module that is a graded version of the regular representation of Sn. Using a straightening algorithm on monomials and the Garsia-Stanton basis, Adin, Brenti, and Roichman gave a description of the Frobenius image of Rn, graded by partitions, in terms of descents of standard Young tableaux. Motivated by the Delta Conjecture of Macdonald polynomials, Haglund, Rhoades, and Shimozono gave an extension of the coinvariant algebra R n,k and an extension of the Garsia-Stanton basis. Chan and Rhoades further extend these results from Sn to the complex reflection group G(r, 1, n) by defining a G(r, 1, n) module S n,k that generalizes the coinvariant algebra for G(r, 1, n). We extend the results of Adin, Brenti, and Roichman to R n,k and S n,k and connect the results for R n,k to skew ribbon tableaux and a crystal structure defined by Benkart et al.

Hall-Littlewood polynomials and a Hecke action on ordered set partitions

“…Adin, Brenti, and Roichman [1] studied a refinement of the classical coinvariant ring R n = R n,n indexed by all possible partitions λ with ≤ n parts which is finer than the degree grading and whose module structure is governed by descent sets of tableaux T ∈ SYT(n). Meyer [12,Thm. 1.4] extended this result to the quotients R n,k for k ≤ n. It may be interesting to refine Corollary 2.6 to obtain a quantum analog of Meyer's results.…”