Murray, John
(2016)
Symmetric Bilinear Forms and Vertices in Characteristic 2.
Working Paper.
arXiv.
Abstract
Let G be a finite group and let k be an algebraically closed field of characteristic 2. Suppose that M is an indecomposable kGmodule which affords a nondegenerate Ginvariant symmetric bilinear form. We assign to M a collection of 2subgroups of G called its symmetric vertices, each of which contains a Green vertex of M with index at most 2. If M is irreducible then its symmetric vertices are uniquely determined, up to Gconjugacy.
If B is the real 2block of G containing M, we show that each symmetric vertex of M is contained in an extended defect group of B. Moreover, we characterise the extended defect groups in terms of symmetric vertices.
In order to prove these results, we develop the theory of involutary Galgebras. This allows us to translate questions about symmetric kGmodules into questions about projective modules of quadratic type.
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