ALGEBRAIC COMBINATORICS

Descent representations for generalized coinvariant algebras
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 805-830.

The coinvariant algebra ${R}_{n}$ is a well-studied ${𝔖}_{n}$-module that is a graded version of the regular representation of ${𝔖}_{n}$. Using a straightening algorithm on monomials and the Garsia–Stanton basis, Adin, Brenti, and Roichman gave a description of the Frobenius image of ${R}_{n}$, 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 Garsia–Stanton basis. Chan and Rhoades further extend these results from ${𝔖}_{n}$ to the complex reflection group $G\left(r,1,n\right)$ by defining a $G\left(r,1,n\right)$ module ${S}_{n,k}$ that generalizes the coinvariant algebra for $G\left(r,1,n\right)$. 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.

Revised: 2020-01-13
Accepted: 2020-01-13
Published online: 2020-08-20
DOI: https://doi.org/10.5802/alco.109
Classification: 05E10,  05E05,  20C30,  05E15
Keywords: Young tableaux, representation theory, descent monomials.
@article{ALCO_2020__3_4_805_0,
author = {Meyer, Kyle P.},
title = {Descent representations for generalized coinvariant algebras},
journal = {Algebraic Combinatorics},
pages = {805--830},
publisher = {MathOA foundation},
volume = {3},
number = {4},
year = {2020},
doi = {10.5802/alco.109},
language = {en},
url = {alco.centre-mersenne.org/item/ALCO_2020__3_4_805_0/}
}
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/item/ALCO_2020__3_4_805_0/

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