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The double Burnside ring via biset functors
Radu Stancu 教授(法国亚棉大学)
2018-01-01 12:13  华东师范大学

曹锡华代数论坛

Abstract

This is a joint work with Serge Bouc and Jacques Thévenaz.

Let G be a finite group and k be afield. The purpose of this talk is to investigate the
structure of the double Burnside ring kB(G;G).

It turns out that the simple kB(G;G)-modules are evaluations at G of simple biset
functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G;H)
and prove that the quotient of the functor kB(-;H) by the radical of the bilinear form is
semi-simple. This allows for a description of the evaluation of simple functors, hence of
simple modules for the double Burnside ring. The evaluation of a simple biset functor at a
finite group G may be zero. We give examples where this happens, as well as where this
does not occur. Under some restrictive conditions on G we can give a closed formula for
such an evaluation.