Descent representations for generalized coinvariant algebras

Meyer, Kyle P.

doi:10.5802/alco.109

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.

Abstract

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 $G (r, 1, n)$ 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.

Received: 2018-02-08
Revised: 2020-01-13
Accepted: 2020-01-13
Published online: 2020-08-20

DOI: 10.5802/alco.109

Classification: 05E10, 05E05, 20C30, 05E15
Keywords: Young tableaux, representation theory, descent monomials.

Author's affiliations:

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/

References
Cited by

[1] Adams, William W.; Loustaunau, Philippe An introduction to Gröbner bases, Graduate Studies in Mathematics, 3, American Mathematical Society, Providence, RI, 1994, xiv+289 pages | DOI | MR | Zbl

[2] Adin, Ron; Brenti, Francesco; Roichman, Yuval Descent representations and multivariate statistics, Trans. Amer. Math. Soc., Volume 357 (2005) no. 8, pp. 3051-3082 | DOI | MR | Zbl

[3] Bagno, Eli; Biagioli, Riccardo Colored-descent representations of complex reflection groups $G (r, p, n)$ , Israel J. Math., Volume 160 (2007) no. 1, pp. 317-347 | DOI | MR | Zbl

[4] Benkart, Georgia; Colmenarejo, Laura; Harris, Pamela E; Orellana, Rosa; Panova, Greta; Schilling, Anne; Yip, Martha A minimaj-preserving crystal on ordered multiset partitions, Adv. in Appl. Math., Volume 95 (2018), pp. 96-115 | DOI | MR | Zbl

[5] Chan, Kin Tung Jonathan; Rhoades, Brendon Generalized coinvariant algebras for wreath products (2017) (https://arxiv.org/abs/1701.06256)

[6] Chevalley, Claude Invariants of finite groups generated by reflections, Amer. J. Math., Volume 77 (1955) no. 4, pp. 778-782 | DOI | MR | Zbl

[7] Garsia, Adriano M Combinatorial methods in the theory of Cohen–Macaulay rings, Adv. in Math., Volume 38 (1980) no. 3, pp. 229-266 | DOI | MR | Zbl

[8] Garsia, Adriano M; Procesi, Claudio On certain graded $S_{n}$ -modules and the $q$ -Kostka polynomials, Adv. Math., Volume 94 (1992) no. 1, pp. 82-138 | DOI | MR | Zbl

[9] Garsia, Adriano M; Stanton, Dennis Group actions on Stanley–Reisner rings and invariants of permutation groups, Adv. in Math., Volume 51 (1984) no. 2, pp. 107-201 | DOI | MR | Zbl

[10] Haglund, James; Rhoades, Brendon; Shimozono, Mark Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture, Adv. Math., Volume 329 (2018), pp. 851-915 | DOI | MR | Zbl

[11] Stanley, Richard P Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 3, pp. 475-511 | DOI | MR | Zbl

[12] Stanley, R.P.; Fomin, S. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR

[13] Stembridge, John On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math., Volume 140 (1989) no. 2, pp. 353-396 | DOI | MR | Zbl

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