## 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 In 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.

Original language | English (US) |
---|---|

Pages (from-to) | 47-80 |

Number of pages | 34 |

Journal | Algebraic Combinatorics |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - 2018 |

## Keywords

- Coinvariant algebra
- Hecke algebra
- Set partition

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics