Representation Theory of $\mathbb{Z}_2^{*n}$

TL;DR

This paper investigates the representation theory of the free product of multiple copies of , analyzing automorphisms, simple components, and geometric invariant theory quotients to understand their structure.

Contribution

It introduces a detailed analysis of the simple representations of ^{*n} using automorphism group actions and studies the smoothness of related GIT-quotients.

Findings

Identification of components containing simple representations

Analysis of local quiver settings for specific representations

Insights into the smoothness of GIT-quotients

Abstract

We study the representations of the group $\mathbb{Z}_2^{*n}$, the free product of $\mathbb{Z}_2$ with itself $n$-times. We use the action of $B_n = S_2 \wr S_n $ as algebra automorphisms on the group algebra $\mathbb{C}(\mathbb{Z}_2^{*n})$ to find the components that contain simple representations and to study smoothness of their GIT-quotients. In particular, all the possible local quiver settings are studied for the component containing the standard $n$-dimensional representation of $S_{n+1}$.