Unitary representations of a group given generating set - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:37:48Z http://mathoverflow.net/feeds/question/116938 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116938/unitary-representations-of-a-group-given-generating-set Unitary representations of a group given generating set jsliyuan 2012-12-21T00:48:11Z 2012-12-21T04:31:53Z <p>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.</p> <p>What are the unitary representations of such group?</p> <p>I only know in the case $G = K_n$ (the complete graph), representation of this group is well studied in Clifford algebra.</p> http://mathoverflow.net/questions/116938/unitary-representations-of-a-group-given-generating-set/116949#116949 Answer by Misha for Unitary representations of a group given generating set Misha 2012-12-21T04:31:53Z 2012-12-21T04:31:53Z <p>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 $R$:</p> <p>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 $G$. </p> <p>One can also prove a scheme-theoretic version of this statement. Proofs are variations on the proof of the main theorem of <a href="http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1998__88_/PMIHES_1998__88__5_0/PMIHES_1998__88__5_0.pdf" rel="nofollow">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.</a> 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. </p>