# ALGEBRAIC COMBINATORICS

Ordered set partitions and the $0$-Hecke algebra
Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 47-80
Let the symmetric group ${𝔖}_{n}$ act on the polynomial ring $ℚ\left[{\mathbf{x}}_{n}\right]=ℚ\left[{x}_{1},\cdots ,{x}_{n}\right]$ by variable permutation. The coinvariant algebra is the graded ${𝔖}_{n}$-module ${R}_{n}:=ℚ\left[{\mathbf{x}}_{n}\right]/{I}_{n}$, where ${I}_{n}$ is the ideal in $ℚ\left[{\mathbf{x}}_{n}\right]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient ${R}_{n,k}$ of the polynomial ring $ℚ\left[{\mathbf{x}}_{n}\right]$ depending on two positive integers $k\le n$ 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 $𝔽\left[{\mathbf{x}}_{n}\right]$ which carries a graded action of the 0-Hecke algebra ${H}_{n}\left(0\right)$, 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.
Revised : 2017-12-07
Accepted : 2017-12-11
Published online : 2018-01-29
DOI : https://doi.org/10.5802/alco.10
Classification:  05E10,  05E15
Keywords: Hecke algebra, set partition, coinvariant algebra
@article{ALCO_2018__1_1_47_0,
author = {Huang, Jia and Rhoades, Brendon},
title = {Ordered set partitions and the $0$-Hecke algebra},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {1},
year = {2018},
pages = {47-80},
doi = {10.5802/alco.10},
zbl = {06882334},
language = {en},
url = {http://alco.centre-mersenne.org/item/ALCO_2018__1_1_47_0}
}

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/item/ALCO_2018__1_1_47_0/

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