The weighted fusion category algebra and the q-Schur algebra for GL2(q)

We show that the weighted fusion category algebra of the principal 2-block b(0) of GL(2)(q) is the quotient of the q-Schur algebra S-2(q) by its socle, for q an odd prime power. As a consequence, we get a canonical bijection between the set of isomorphism classes of simple kGL(2) (q)b(0)-modules and the set of conjugacy classes of b(0)-weights in this case. (C) 2007 Elsevier Inc. All rights reserved.