Author: John Murray

Title: Projective indecomposable modules, Scott modules and the Frobenius-Schur indicator

Abstract:

Let $\Phi$ be a principal indecomposable character of a finite group $G$ in characteristic $2$. The Frobenius-Schur indicator $\nu(\Phi)$ of $\Phi$ is shown to equal the rank of a bilinear form defined on the span of the involutions in $G$. Moreover, if the principal indecomposable module corresponding to $\Phi$ affords a quadratic geometry, then $\nu(\Phi)>0$. This result is used to prove a more precise form of a theorem of Benson and Carlson on the existence of Scott components in the endomorphism ring of an indecomposable $G$-module, in case the module affords a $G$-invariant symmetric form.

Status: submitted for publication