Brauer's Main Theorems

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• Stacey Law, University of Cambridge
• Friday 05 February 2016, 15:00-16:00
• CMS, MR4.

If you have a question about this talk, please contact Nicolas Dupré.

Brauer’s Main Theorems are results in the modular representation theory of finite groups that link the blocks of a finite group G with those of its p-local subgroups. Over characteristic not dividing the group order, all finite-dimensional modules are projective and the group algebra is semisimple. This unsurprisingly does not hold for a field k of characteristic p dividing |G|, and we will introduce certain p-subgroups Q of G called vertices as measures of ‘how far from projective’ modules are, then extend this into the concept of defect groups D for the blocks of the group algebra. We will see that kG-module structure can be related to that of N_G(Q) and N_G(D) using the Green correspondence and Brauer’s Main Theorems, through small concrete examples as well as theoretical applications. If there’s time we’ll also outline Brauer-Dade theory for cyclic blocks, where the simples and indecomposable projectives can be described neatly using graphs known as Brauer trees.

This talk is part of the Junior Algebra and Number Theory seminar series.

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