Ordered set partitions and the 0-Hecke algebra

Jia Huang, Brendon Rhoades

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2 Scopus citations


Let the symmetric group Snact 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 Rn,kof the polynomial ring Q[xn] depending on two positive integers k ≤ n which reduces to the classical coinvariant algebra of the symmetric group Snwhen k = n. The quotient Rn,kcarries 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 Sn,kof 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.

Original languageEnglish (US)
Pages (from-to)47-80
Number of pages34
JournalAlgebraic Combinatorics
Issue number1
StatePublished - 2018


  • Coinvariant algebra
  • Hecke algebra
  • Set partition

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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