Clifford theory for glider representations

TL;DR

This paper extends classical Clifford theory to chains of normal subgroups, introducing the concept of glider representations and irreducibility for a more nuanced understanding of group module decompositions.

Contribution

It develops a new framework for Clifford theory involving chains of normal subgroups and introduces irreducibility of fragments, broadening the scope of representation analysis.

Findings

Introduces glider representations for chains of groups.

Defines irreducibility for fragments in this context.

Constructs induced fragment structures for normal subchains.

Abstract

Classical Clifford theory studies the decomposition of simple $G$-modules into simple $H$-modules for some normal subgroup $H \triangleleft G$. In this paper we deal with chains of normal subgroups $1 \triangleleft G_1 \triangleleft \cdots \triangleleft G_d =G$, which allow to consider fragments and in particular glider representations. These are given by a descending chain of vector spaces over some field $K$ and relate different representations of the groups appearing in the chain. Picking some normal subgroup $H \triangleleft G$ one obtains a normal subchain and one can construct an induced fragment structure. Moreover, a notion of irreducibility of fragments is introduced, which completes the list of ingredients to perform a Clifford theory.