Ordered set partitions and the $0$-Hecke algebra

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.