Ordered set partitions and the $0$-Hecke algebra

Huang, Jia; Rhoades, Brendon

doi:10.5802/alco.10

Ordered set partitions and the

0

-Hecke algebra

Huang, Jia ¹; Rhoades, Brendon ²

¹ University of Nebraska at Kearney Department of Mathematics Kearney, NE, 68849 (USA)
² University of California, San Diego Department of Mathematics La Jolla, CA, 92093 (USA)

Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 47-80.

Abstract

Let the symmetric group $𝔖_{n}$ act on the polynomial ring $ℚ [x_{n}] = ℚ [x_{1}, \dots, x_{n}]$ by variable permutation. The coinvariant algebra is the graded $𝔖_{n}$ -module $R_{n} : = ℚ [x_{n}] / I_{n}$ , where $I_{n}$ is the ideal in $ℚ [x_{n}]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n, k}$ of the polynomial ring $ℚ [x_{n}]$ depending on two positive integers $k \leq n$ which reduces to the classical coinvariant algebra of the symmetric group $𝔖_{n}$ when $k = n$ . The quotient $R_{n, k}$ carries the structure of a graded $𝔖_{n}$ -module; Haglund et. al. determine its graded isomorphism type and relate it to the Delta Conjecture in the theory of Macdonald polynomials. We introduce and study a related quotient $S_{n, k}$ of $𝔽 [x_{n}]$ which carries a graded action of the 0-Hecke algebra $H_{n} (0)$ , where $𝔽$ is an arbitrary field. We prove 0-Hecke analogs of the results of Haglund, Rhoades, and Shimozono. In the classical case $k = n$ , we recover earlier results of Huang concerning the 0-Hecke action on the coinvariant algebra.

Received: 2017-08-09
Revised: 2017-12-07
Accepted: 2017-12-11
Published online: 2018-01-29

MR Zbl

DOI: 10.5802/alco.10

Classification: 05E10, 05E15
Keywords: Hecke algebra, set partition, coinvariant algebra

Author's affiliations:

Huang, Jia ¹; Rhoades, Brendon ²

¹ University of Nebraska at Kearney Department of Mathematics Kearney, NE, 68849 (USA)
² University of California, San Diego Department of Mathematics La Jolla, CA, 92093 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Huang, Jia; Rhoades, Brendon. Ordered set partitions and the $0$-Hecke algebra. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 47-80. doi : 10.5802/alco.10. https://alco.centre-mersenne.org/articles/10.5802/alco.10/

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