Hyperbolic Modules of Finite Group Algebras over Finite Fields of Characteristic Two

TL;DR

This paper characterizes hyperbolic modules over finite group algebras in characteristic two using $F$-special subgroups and elements, with implications for characters, codes, and Witt groups.

Contribution

It introduces the concept of $F$-special subgroups and elements and provides criteria for hyperbolicity of modules based on these, extending understanding in modular representation theory.

Findings

Hyperbolic modules are characterized by restrictions to $F$-special subgroups.

Characteristic polynomials of $F$-special elements are squares of polynomials over $F$.

Applications include insights into characters, self-dual codes, and Witt groups.

Abstract

Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. We introduce \emph{$F$-special subgroups} and \emph{$F$-special elements} of $G$. In the case where $F$ contains a $p$th primitive root of unity for each odd prime $p$ dividing the order of $G$ (e.g. it is the case once $F$ is a splitting field for all subgroups of $G$), the $F$-special elements of $G$ coincide with real elements of odd order. We prove that a symmetric $FG$-module $V$ is hyperbolic if and only if the restriction $V_D$ of $V$ to every $F$-special subgroup $D$ of $G$ is hyperbolic, and also, if and only if the characteristic polynomial on $V$ defined by every $F$-special element of $G$ is a square of a polynomial over $F$. Some immediate applications to characters, self-dual codes and Witt groups are given.