Generalized characters for glider representations of groups

TL;DR

This paper extends character theory to glider representations associated with group filtrations, providing new tools for analyzing both abelian and non-abelian groups.

Contribution

It develops a generalized character theory for glider representations, including Artin's theorem generalization and explicit calculations for abelian groups.

Findings

Generalized character ring computed for finite abelian groups

Semisimple quotient of the generalized character ring analyzed

Discussion of quaternion group as a non-abelian example

Abstract

Glider representations can be defined for a finite algebra filtration FKG determined by a chain of subgroups 1 < G_1 < ... < G_d = G. In this paper we develop the generalized character theory for such glider representations. We give the generalization of Artin's theorem and define a generalized inproduct. For finite abelian groups G with chain 1 < G, we explicitly calculate the generalized character ring and compute its semisimple quotient. The papers ends with a discussion of the quaternion group as a first non-abelian example.