Specht modules and Kazhdan–Lusztig cells in type BnBn

Dipper, James and Murphy generalized the classical Specht module theory to the Hecke algebras of type BnBn. On the other hand, for any choice of a monomial order on the parameters of type BnBn, we obtain the corresponding Kazhdan–Lusztig cell modules. In this paper, we show that the Specht modules are naturally isomorphic to the Kazhdan–Lusztig cell modules if we choose the dominance order on the parameters, as in the “asymptotic case” studied by Bonnafé and the second named author. We also give examples which show that such an isomorphism does not exist for other choices of monomial orders.