Glider Representation Rings with a view on distinguishing groups

TL;DR

This paper develops the theory of glider representation rings for finite groups, providing new algebraic decompositions and applications to distinguishing isocategorical groups.

Contribution

It introduces a detailed framework for glider representation rings and establishes a Wedderburn-Malcev type decomposition relating to group representation theory.

Findings

Decomposition of the glider representation ring as a subring of the Grothendieck ring.

A module decomposition of the associated $Q$-algebra relating to subnormal subgroups.

Complete algebra decomposition for nilpotent groups of class 2.

Abstract

Let $G$ be a finite group. In the first part of the paper we develop further the foundations of the youngly introduced glider representation theory. Glider representations encompass filtered modules over filtered rings and as such carry much information of $G$. Therefore the main focus is on the glider representation ring $R_d(\widetilde{G})$, which is shown to be realisable as a concrete subring of the split Grothendieck ring of the monoidal category $\text{glid}_d(G)$ of (Noetherian) glider $\mathbb{C}$-representations of (length $d$) of $G$. In the second part we investigate a Wedderburn-Malcev type decomposition of the (infinite-dimensional) $\mathbb{Q}$-algebra $\mathbb{Q}(\widetilde{G}) := \mathbb{Q} \otimes_{\mathbb{Z}}R_1(\widetilde{G})$. The main theorem obtains a $\mathbb{Q}[G^{ab}]$-module decomposition of $\mathbb{Q}(\widetilde{G})$ relating it in a precise way to $\mathbb{C}$-representation theory of subnormal subgroups in $G$. Under certain vanishing assumptions, which are proven to hold for nilpotent groups (of class $2$), the second main theorem completely describes a $\mathbb{Q}[G^{ab}]$-algebra decomposition. We end with pointing out applications on distinguishing isocategorical groups.