Ordered set partitions and the $0$-Hecke algebra

Jia Huang,Brendon Rhoades

doi:10.5802/alco.10

Abstract

Let the symmetric group 𝔖 n act on the polynomial ring ℚ[x n ]=ℚ[x 1 ,⋯,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≤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.

Highlights

The purpose of this paper is to define and study a 0-Hecke analog of a recently defined graded module for the symmetric group [16]
The symmetric group Sn acts on the polynomial ring Q[xn] := Q[x1, . . . , xn] by variable permutation
The corresponding invariant subring Q[xn]Sn consists of all f ∈ Q[xn] with w(f ) = f for all w ∈ Sn, and is generated by the elementary symmetric functions e1(xn), . . . , en(xn), where ed(xn) = ed(x1, . . . , xn) :=

Summary

Introduction

The purpose of this paper is to define and study a 0-Hecke analog of a recently defined graded module for the symmetric group [16]. Proving 0-Hecke analogs of module theoretic results concerning the symmetric group has received a great deal of recent study in algebraic combinatorics [4, 17, 18, 26]; let us recall the 0-Hecke analog of the variable permutation action of Sn on a polynomial ring. We have a 0-Hecke analog of the permutation action of Sn on OPn,k It is well-known that the 0-Hecke algebra Hn(0) has another generating set {π1, . For k < n the ideal In,k is not usually stable under the action of Hn(0) on F[xn], so that the quotient ring Rn,k = F[xn]/In,k does not have the structure of an Hn(0)-module.

Background

Hilbert Series and Artin basis

Garsia-Stanton type bases

Module structure over the 0-Hecke algebra

Characteristic formulas

Conclusion