On tensor spaces over Hecke algebras of type B_n

Let W (B_n) be the Weyl group of type B_n and H (B_n) be the associated Iwahori-Hecke algebra. In this paper, we study the n-tensor space V^{⊗ n} (where dim V = 2 m) with natural actions (introduced in [R.M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997) 418-438]) of W (B_n) and of H (B_n). For each composition λ = (λ₁, ..., λ_m) of n, let e_λ 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 S_λ in W (B_n), then the action of the natural basis element T_d on e_λ 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 (B_n)) with respect to certain standard parabolic subalgebras.