Ordered set partitions and the $0$-Hecke algebra

Huang, Jia; Rhoades, Brendon

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

Let the symmetric group ${𝔖}_n$ act on the polynomial ring $\mathbb{Q}\left[{\mathbf{x}}_n\right]=\mathbb{Q}[x_1,\cdots ,x_n]$ by variable permutation. The coinvariant algebra is the graded ${𝔖}_n$-module $R_n:=\mathbb{Q}\left[{\mathbf{x}}_n\right]/I_n$, where $I_n$ is the ideal in $\mathbb{Q}\left[{\mathbf{x}}_n\right]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n,k}$ of the polynomial ring $\mathbb{Q}\left[{\mathbf{x}}_n\right]$ 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.

Classification : 05E10, 05E15
Mots-clés : Hecke algebra, set partition, coinvariant algebra

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