0-Hecke algebra actions on coinvariants and flags

Jia Huang

doi:10.1007/s10801-013-0486-1

0-Hecke algebra actions on coinvariants and flags

Jia Huang

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

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	https://doi.org/10.1007/s10801-013-0486-1
State	Published - Aug 2014
Externally published	Yes

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

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10.1007/s10801-013-0486-1

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title = "0-Hecke algebra actions on coinvariants and flags",

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.",

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

author = "Jia Huang",

note = "Funding Information: Acknowledgements The author is grateful to Victor Reiner for asking questions which led to the research, for giving many helpful suggestions, and for his support from NSF grant DMS-1001933. He is indebted to Andrew Berget, Tom Denton, Anne Schilling, and Monica Vazirani for interesting conversations and hospitality during the visit to UC Davis. He thanks Marcelo Aguiar, Sami Assaf, Nantel Bergeron, Adriano Garsia, Patricia Hersh, Liping Li, Sarah Mason, Dennis Stanton and Peter Webb for their helpful suggestions and comments.",

year = "2014",

month = aug,

doi = "10.1007/s10801-013-0486-1",

language = "English (US)",

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journal = "Journal of Algebraic Combinatorics",

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AU - Huang, Jia

N1 - Funding Information: Acknowledgements The author is grateful to Victor Reiner for asking questions which led to the research, for giving many helpful suggestions, and for his support from NSF grant DMS-1001933. He is indebted to Andrew Berget, Tom Denton, Anne Schilling, and Monica Vazirani for interesting conversations and hospitality during the visit to UC Davis. He thanks Marcelo Aguiar, Sami Assaf, Nantel Bergeron, Adriano Garsia, Patricia Hersh, Liping Li, Sarah Mason, Dennis Stanton and Peter Webb for their helpful suggestions and comments.

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

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JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

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ER -

0-Hecke algebra actions on coinvariants and flags

Abstract

Keywords

ASJC Scopus subject areas

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