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

Let n >= 4 be ail even integer. Let K be a field with char K not equal 2 and q an invertible element in K such that Pi(n-1)(i=1) (1 + q(i)) not equal 0. In this paper, we study the decomposition numbers over K of the Iwahori-Hecke algebra H-q(D-n) of type D-n 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 char K = 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 H-q(D-n) 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. (C) 2008 Elsevier Inc. All rights reserved.