Abstract
The Hecke algebra of the pair $(\mathcal{S}_{2n},\mathcal{B}_n)$, where $\mathcal{B}_n$ is the hyperoctahedral subgroup of $\mathcal{S}_{2n}$, was introduced by James in 1961. It is a natural analogue of the center of the symmetric group algebra. In this paper, we give a polynomiality property of its structure coefficients. Our main tool is a combinatorial algebra which projects onto the Hecke algebra of $(\mathcal{S}_{2n},\mathcal{B}_n)$ for every $n$. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.
Highlights
For a positive integer n, let Sn denote the symmetric group of permutations on the set [n] := {1, 2, · · ·, n}, and let C[Sn] denote the group-algebra of Sn over C1, the field of complex numbers
We introduce the notion of trivial extension of a partial bijection of [2n] and we use it the electronic journal of combinatorics 21(4) (2014), #P4.35 to build a homomorphism between the partial bijection algebra of n and the symmetric group algebra of 2n
In [IK99, Section 9], Ivanov and Kerov have given an isomorphism between the algebra of 1-shifted symmetric functions and the algebra A∞, which is the universal algebra that projects on the center of the symmetric group algebra Z(C[Sn]), for each n
Summary
For a positive integer n, let Sn denote the symmetric group of permutations on the set [n] := {1, 2, · · · , n}, and let C[Sn] denote the group-algebra of Sn over C1, the field of complex numbers. In [IK99], by using objects called partial permutations, the same result is obtained by Ivanov and Kerov This more recent proof provides a combinatorial description of the coefficients of the relevant polynomials. In this paper we establish a polynomiality property for the structure coefficients of the Hecke algebra of (S2n, Bn). Méliot has used this same idea in [Mél13] to give a polynomiality property for the structure coefficients of the center of the group-algebra C[GL(n, Fq)], where GL(n, Fq) is the group of invertible n×n matrices with coefficients in Fq. Because of the similarities between Meliot’s construction and ours, we are convinced we should build a general framework in which such a result (polynomiality of the structure coefficients) always holds.
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