ALGEBRAIC COMBINATORICS

no. 4
Descent representations for generalized coinvariant algebras
Meyer, Kyle P.1
1 Dept. of Mathematics University of California, San Diego 9500 Gilman Dr. La Jolla CA 92093, USA
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 805-830.

The coinvariant algebraĀ R n is a well-studied š¯”– n -module that is a graded version of the regular representation of š¯”– n . Using a straightening algorithm on monomials and the Garsiaā€“Stanton basis, Adin, Brenti, and Roichman gave a description of the Frobenius image ofĀ R n , 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 Garsiaā€“Stanton basis. Chan and Rhoades further extend these results from š¯”– n 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.

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DOI: 10.5802/alco.109
Classification: 05E10, 05E05, 20C30, 05E15
Keywords: Young tableaux, representation theory, descent monomials.
Meyer, Kyle P. 1

1 Dept. of Mathematics University of California, San Diego 9500 Gilman Dr. La Jolla CA 92093, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Meyer, Kyle P. Descent representations for generalized coinvariant algebras. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 805-830. doi : 10.5802/alco.109. https://alco.centre-mersenne.org/articles/10.5802/alco.109/

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