On tensor spaces over Hecke algebras of type Bn

Let W(Bn) be the Weyl group of type Bn and H(Bn) be the associated Iwahori–Hecke algebra. In this paper, we study the n-tensor space V⊗n (where dimV=2m) with natural actions (introduced in [R.M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997) 418–438]) of W(Bn) and of H(Bn). For each composition λ=(λ1,…,λm) of n, let eλ be the corresponding initial basis element of V⊗n (see (3.8) for definition). We show that, if d is a distinguished right coset representative of Sλ in W(Bn), then the action of the natural basis element Td on eλ coincides with the * permutation action of d up to a scalar. As an application, we prove that the n-tensor space decomposes (at the integral level) into a direct sum of some permutation modules (over Hecke algebra H(Bn)) with respect to certain standard parabolic subalgebras.