Structure Coefficients of the Hecke Algebra of (S_2n, B_n)

The Hecke algebra of the pair (S_2n, B_n), where B_n is the hyperoctahedral subgroup of 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 (S_2n, B_n) for every n. To build it, by using partial bijections we introduce and study a new class of finite dimensional algebras.