Mathematics Colloquia and Seminars

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The coinvariant algebra $R_n$ is a well-studied $\mathfrak{S}_n$-module that gives a graded version of the regular representation of $\mathfrak{S}_n$. Motivated by the Delta Conjecture in the field of Macdonald polynomials, Haglund, Rhoades, and Shimozono define a graded algebra and $\mathfrak{S}_n$-module $R_{n,k}$ that generalizes the coinvariant algebra. For the classical coinvariant algebra, Adin, Brenti, and Roichman give a refinement of the grading that is indexed by partitions and whose isomorphism types are determined by descents of standard Young tableaux.

In this talk I will cover these algebras and give an extension of the results of Adin, Brenti, and Roichman to $R_{n,k}$. Additionally I will discus how this extension relates to work of Benkart, Colmenarejo, Harris, Orellano, Panova, Schilling, and Yip on a minimaj crystal and skew ribbon tableaux; I will also discuss using a method of Garsia and Procesi to find further generalizations.