Computing singularities of the spectra of representation rings of finite groups

TL;DR

This paper investigates the singularities and tangent space dimensions of the spectra of representation rings of finite groups, providing explicit computations for small groups and methods for general cases.

Contribution

It introduces practical methods to compute singularities and tangent space dimensions of spectra of representation rings for any finite group, extending previous theoretical work.

Findings

Explicit singularity computations for small noncommutative groups

Development of practical algorithms for arbitrary finite groups

Analysis of the fibers of the formal tangent sheaf

Abstract

Let $G$ be a finite group of order $n$, and $\xi$ an $n$-th primitive root of unity. Consider the affine scheme $C:=\mbox{Spc}({\mathbb Z}[\xi]\otimes_{\mathbb Z} R(G))$ where $R(G)$ is the representation ring of $G$. We study the fibers of the formal tangent sheaf of $C$ by computing their dimension and also finding (and measuring) the singularities of $C$. We present explicit computations for noncommutative groups of small order, and develop practical methods to compute these invariants for an arbitrary finite group.