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

Abstract

We study the representations of the group ${\mathbb{Z}}_2^{\ast n}$, 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 $\mathbb{C}({\mathbb{Z}}_2^{\ast n})$ 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 $S_{n+1}$.

31 pages, 6 figures. Comments welcome