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Journal of the London Mathematical Society 2001 63(2):336-352; doi:10.1017/S0024610700001873
© 2001 by London Mathematical Society
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© The London Mathematical Society

Kazhdan–Lusztig Cells, q-Schur Algebras and James' Conjecture

Meinolf Geck

Institut Girard Desargues Bâtiment 101, Université Lyon 1 43 boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France, geck{at}desargues.univ-lyon1.fr

Received 4 October 1999. Revision received 5 April 2000.

We consider the Dipper–James q-Schur algebra Sq(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 Sq(n, r)k is a semisimple algebra if q is not a root of unity. Much more interesting is the case when Sq(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.


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