Andrew Mathas (University of Sydney)
Friday 26th May, 12.05-12.55pm, Carslaw 159
Seminormal forms for cellular algebras
Let A be a split semisimple algebra. Then, by the Wedderburn Theorem, A is isomorphic to a direct sum of matrix rings. It is natural to ask whether we can find an explicit basis for A which realizes its Wedderburn decomposition. We may assume that A is cellular, in the sense of Graham and Lehrer, since every split semisimple algebra is cellular. We show how to construct an explicit Wedderburn basis of any cellular algebra which has a family of "JM-elements". In fact, every split semisimple algebra has a family of JM-elements and we can give examples for the most interesting cellular algebras. Surprisingly this theory has strong implications for non-semisimple cellular algebras. In particular, we obtain an explicit basis for the blocks of several algebras, including the group algebras of the symmetric groups over arbitrary fields.