KAZHDAN–LUSZTIG CELLS, q-SCHUR ALGEBRAS AND JAMES' CONJECTURE

Abstract

We consider the Dipper–James q-Schur algebra [Sscr ]_q(n, r)_k, defined over a field k and with parameter q ≠ 0. An understanding of the representation theory of this algebra is of considerable interest in the representation theory of finite groups of Lie type and quantum groups; see, for example, [6] and [11]. It is known that [Sscr ]_q(n, r)_k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when [Sscr ]_q(n, r)_k is not semisimple. Then we have a corresponding decomposition matrix which records the multiplicities of the simple modules in certain ‘standard modules’ (or ‘Weyl modules’). A major unsolved problem is the explicit determination of these decomposition matrices.