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.

  • $\begingroup$ It's unclear whether you're talking about a group or an algebra. $\endgroup$ Dec 21 '12 at 1:00
  • $\begingroup$ It's a group. Sorry, I should have added $1$ and $-1$ in the set of generators. $\endgroup$
    – jsliyuan
    Dec 21 '12 at 1:05
  • 1
    $\begingroup$ 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). $\endgroup$ 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 $G$.

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. 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.


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