### Abstract

The 0-Hecke algebra H _{n} (0) is a deformation of the group algebra of the symmetric group Sn. We show that its coinvariant algebra naturally carries the regular representation of H _{n} (0), giving an analogue of the well-known result for Sn by Chevalley-Shephard-Todd. By investigating the action of H _{n} (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H _{n} (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall-Littlewood symmetric functions indexed by hook shapes.

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

Pages (from-to) | 245-278 |

Number of pages | 34 |

Journal | Journal of Algebraic Combinatorics |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2014 |

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### Keywords

- 0-Hecke algebra
- Coinvariant algebra
- Demazure operator
- Descent monomial
- Flag variety
- Hall-Littlewood function
- Ribbon number
- Springer fiber

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

### Cite this

**0-Hecke algebra actions on coinvariants and flags.** / Huang, Jia.

Research output: Contribution to journal › Article

*Journal of Algebraic Combinatorics*, vol. 40, no. 1, pp. 245-278. https://doi.org/10.1007/s10801-013-0486-1

}

TY - JOUR

T1 - 0-Hecke algebra actions on coinvariants and flags

AU - Huang, Jia

PY - 2014/8

Y1 - 2014/8

N2 - The 0-Hecke algebra H n (0) is a deformation of the group algebra of the symmetric group Sn. We show that its coinvariant algebra naturally carries the regular representation of H n (0), giving an analogue of the well-known result for Sn by Chevalley-Shephard-Todd. By investigating the action of H n (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H n (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall-Littlewood symmetric functions indexed by hook shapes.

AB - The 0-Hecke algebra H n (0) is a deformation of the group algebra of the symmetric group Sn. We show that its coinvariant algebra naturally carries the regular representation of H n (0), giving an analogue of the well-known result for Sn by Chevalley-Shephard-Todd. By investigating the action of H n (0) on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of H n (0) on the cohomology rings of the Springer fibers, and similarly interpret the (non-commutative) Hall-Littlewood symmetric functions indexed by hook shapes.

KW - 0-Hecke algebra

KW - Coinvariant algebra

KW - Demazure operator

KW - Descent monomial

KW - Flag variety

KW - Hall-Littlewood function

KW - Ribbon number

KW - Springer fiber

UR - http://www.scopus.com/inward/record.url?scp=84904767222&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904767222&partnerID=8YFLogxK

U2 - 10.1007/s10801-013-0486-1

DO - 10.1007/s10801-013-0486-1

M3 - Article

AN - SCOPUS:84904767222

VL - 40

SP - 245

EP - 278

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 1

ER -