Symmetric bilinear forms and vertices in characteristic 2

TL;DR

This paper introduces symmetric vertices for modules over finite groups in characteristic 2, linking them to extended defect groups and developing involutary G-algebras to analyze symmetric bilinear forms.

Contribution

It defines symmetric vertices in characteristic 2, relates them to Green vertices and extended defect groups, and develops involutary G-algebras for module analysis.

Findings

Symmetric vertices contain Green vertices with index at most 2.

Symmetric vertices are determined up to conjugacy for irreducible modules.

Symmetric vertices are contained in extended defect groups and characterize these groups.

Abstract

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $2$ and let $M$ be an indecomposable $kG$-module which affords a non-degenerate $G$-invariant symmetric bilinear form. We introduce the symmetric vertices of $M$. Each of these is a $2$-subgroup of $G$ which contains a Green vertex of $M$ with index at most $2$. If $M$ is irreducible then its symmetric vertices are determined up to $G$-conjugacy. If $B$ is the real $2$-block 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 $G$-algebras. This allows us to translate questions about symmetric $kG$-modules into questions about projective modules of quadratic type.