Structure coefficients of the Hecke algebra of \((\mathcal{S}_{2n},\mathcal{B}_n)\) (Q470965)

Summary: 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 \textit{A. T. James} [Ann. Math. (2) 74, 456--469 (1961; Zbl 0104.02803)]. 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.