# Unitary representations of a group given generating set

A group $G$ is generated by $1, -1, g_1, g_2, \ldots, g_n$. The relation of its generators is given by a simple undirected graph $G = (V=[n], E)$, where $(i, j) \in E$ means $g_i g_j = -g_j g_i$. In the group $1$ is the identity such that $1g = g$ for any element $g$; $(-1)*(-1) = 1$, and $(-1)$ is commuting with all elements.

What are the unitary representations of such group?

I only know in the case $G = K_n$ (the complete graph), representation of this group is well studied in Clifford algebra.

• It's unclear whether you're talking about a group or an algebra. – Qiaochu Yuan Dec 21 '12 at 1:00
• It's a group. Sorry, I should have added $1$ and $-1$ in the set of generators. – jsliyuan Dec 21 '12 at 1:05
• Representations of this group when $G = K_n$ are not the same as representations of the corresponding Clifford algebra (you need the extra condition that the generator called $-1$ actually acts as $-1$ on the representation). – Qiaochu Yuan Dec 21 '12 at 1:09

Groups that you described are central extensions of RAAGs (Right Angled Artin Groups). Let's call such groups "almost RAAGs" for lack of a better name (since the name "extended Artin groups" is already taken by Looijenga). The answer to your question (for a general graph $G$) is: "Awfully complicated." For instance, consider the space of representations $R=R(G)=Hom(\Gamma, U(2))$ of an almost RAAG $\Gamma=\Gamma_G$. This space has natural structure of an affine real-algebraic set defined by polynomial equations with integer coefficients. The following theorem says that "morally speaking" these are the only (local) restrictions on $R$:
Theorem. Let $X$ be any affine real-algebraic set defined over ${\mathbb Z}$. Then there exists a natural number $n$, a finite graph $G$ and a representation $\rho: \Gamma_G\to U(2)$ so that the germ of $R(G)$ at $\rho$ is isomorphic to an open subset of $X\times {\mathbb R}^n$ containing $0$. In particular, given any closed smooth manifold $M$, there exist $n$ and $G$ so that $M\times {\mathbb R}^n$ embeds as an open subset in $Hom(\Gamma_G, U(2))$ for some $G$.