TY - JOUR

T1 - Hecke algebras with independent parameters

AU - Huang, Jia

N1 - Funding Information:
The author is grateful to Pasha Pylyavskyy and Victor Reiner for asking inspiring questions which lead to this work. He thanks the anonymous referee for helpful suggestions and Victor Reiner for partial support from NSF Grant DMS-1001933.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2016/5/1

Y1 - 2016/5/1

N2 - We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

AB - We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) is commutative, which happens only if the Coxeter diagram of (W, S) is simply laced and bipartite. In particular, for type A, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

KW - Fibonacci number

KW - Grothendieck group

KW - Hecke algebra

KW - Independent parameters

KW - Independent set

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U2 - 10.1007/s10801-015-0645-7

DO - 10.1007/s10801-015-0645-7

M3 - Article

AN - SCOPUS:84946135164

VL - 43

SP - 521

EP - 551

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

IS - 3

ER -