Séminaire Lotharingien de Combinatoire, 82B.56 (2019), 12 pp.
Kyle P. Meyer
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(2005) define modules Rn,μ, that refine the grading of Rn, and they describe the Frobenius image of Rn,μ in terms of standard Young tableaux with certain descents. Motivated by the Delta Conjecture of Macdonald polynomials, Haglund, Rhoades, and Shimozono (2016) define a module Rn,k that extends the coinvariant algebra. Also motivated by the Delta Conjecture, Benkart et al.~(2018) defined a crystal structure in terms of the minimaj statistic that up to some twisting has character equal to the Frobenius image of Rn,k. We generalize the results of Adin, Brenti, and Roichman by defining modules Rn,k,μ that refine Rn,k and give a combinatorial description of the Frobenius image. This description not only refines and simplifies some of the results of Haglund, Rhoades, and Shimozono, but also gives a simpler method of obtaining their results. Additionally, these modules give a representation theoretic interpretation for the characters of crystals that Benkart et al. use to build up their minimaj crystal.
Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.
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