Glider Representation Rings with a view on distinguishing groups

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.