Ordered set partitions and the 0-Hecke algebra

Abstract

Let the symmetric group ${\mathfrak{S}}_n$ act on the polynomial ring $\mathbb{Q}[{\mathbf{x}}_n]=\mathbb{Q}[x_1,\dots ,x_n]$ by variable permutation. The coinvariant algebra is the graded ${\mathfrak{S}}_n$-module $R_n:=\mathbb{Q}[{\mathbf{x}}_n]/I_n$, where $I_n$ is the ideal in $\mathbb{Q}[{\mathbf{x}}_n]$ generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient $R_{n,k}$ of the polynomial ring $\mathbb{Q}[{\mathbf{x}}_n]$ depending on two positive integers $k\le n$ which reduces to the classical coinvariant algebra of the symmetric group ${\mathfrak{S}}_n$ when $k=n$. The quotient $R_{n,k}$ carries the structure of a graded ${\mathfrak{S}}_n$-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 $\mathbb{F}[{\mathbf{x}}_n]$ which carries a graded action of the 0-Hecke algebra $H_n(0)$, where $\mathbb{F}$ 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.

30 pages. Corrected the bijection following Definition 3.2

References in corpus (1)

• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

Cited by in corpus (2)

• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
• Representation stability for sequences of 0-Hecke modules