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.



Unitary representations of a group given generating set
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 $G$. 


One can also prove a scheme-theoretic version of this statement. Proofs are variations on the proof of the main theorem of [M.Kapovich, J.J.Millson, "On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex-algebraic varieties", Math. Publications of IHES, vol. 88 (1999), p. 5-95.](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1998__88_/PMIHES_1998__88__5_0/PMIHES_1998__88__5_0.pdf) The idea is that one can encode an arbitrary system of polynomial equations with integer coefficients into (2-dimensional) unitary representation theory of some RAAG (or an almost RAAG), which, in turn, is a variation on the proof of Mnev's Universality Theorem. 


