Ordered set partitions and the $0$-Hecke algebra

Abstract

Let the symmetric group Sn act on the polynomial ring Q[xn] = Q[x1, . . . , xn] by variable permutation. The coinvariant algebra is the graded Sn-module Rn := Q[xn]/In, where In is the ideal in Q[xn] generated by invariant polynomials with vanishing constant term. Haglund, Rhoades, and Shimozono introduced a new quotient R n,k of the polynomial ring Q[xn] depending on two positive integers k ≤ n which reduces to the classical coinvariant algebra of the symmetric group Sn when k = n. The quotient R n,k carries the structure of a graded Sn-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[xn] which carries a graded action of the 0-Hecke algebra Hn (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.

Hall-Littlewood polynomials and a Hecke action on ordered set partitions

“…. , e n−k+1 (x n ) is the ideal J n,k studied by Huang-Rhoades in the context of the 0-Hecke algebra H n (0) [10].…”

“…In the work of Huang-Rhoades on an action of the 0-Hecke algebra on ordered set partitions [10], the acting algebraic object was not a group but rather the 0-Hecke algebra H n (0). Despite this, in [10] it is proven that if we let Y be the point set…”

“…When q = 0, the point set Y (0) n,k becomes 'degenerate' when k < n; there are fewer points in Y (0) n,k than there are ordered set partitions in OP n,k . The point set in [10] used to study the 0-Hecke structure of R n,k and in this way is a quantum deformation of the point set used in [9].…”

“…, x i , the proof of [10, Lem. 4.2] (see also [10,Cor. 4.3]) goes through directly to show that the generalized GS monomials…”

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

“…It is intimately related with the Hopf algebras of quasisymmetric functions and noncommutative symmetric functions respectively [13], in the same way as symmetric group representation theory is intimately connected with the Hopf algebra of symmetric functions. More information about H n (0) and its representations can be found in [10,34], and contemporary results can be found in [6,23,24,25,26,29]. Our interest in the 0-Hecke algebra stems from the authors' previous work in the context of providing a representation-theoretic interpretation for quasisymmetric Schur functions [45].…”