On the decomposition numbers of the Hecke algebra of type Dn when n is even

Let n⩾4 be an even integer. Let K be a field with charK≠2 and q an invertible element in K such that . In this paper, we study the decomposition numbers over K of the Iwahori–Hecke algebra Hq(Dn) of type Dn. We obtain some equalities which relate its decomposition numbers with certain Schur elements and the decomposition numbers of various Iwahori–Hecke algebras of type A with the same parameter q. When charK=0, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence theorem established in [J. Hu, A Morita equivalence theorem for Hecke algebra Hq(Dn) when n is even, Manuscripta Math. 108 (2002) 409–430] and certain twining character formulae of Weyl modules over a tensor product of two q-Schur algebras.