Finite subgroups of Spin(9) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:07:44Z http://mathoverflow.net/feeds/question/102154 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102154/finite-subgroups-of-spin9 Finite subgroups of Spin(9) Maurizio Parton 2012-07-13T16:43:53Z 2012-08-03T15:00:35Z <p>I'm trying to classify compact manifolds $M^{16}$ with a metric which is locally conformal to a (local) metric with holonomy (included in) Spin(9)$\subset$SO(16). To do this, I would need a complete list of finite subgroups of Spin(9) acting freely on $S^{15}$. Any hint on finite subgroups of Spin(9) (even not acting freely) would be very helpful!</p> http://mathoverflow.net/questions/102154/finite-subgroups-of-spin9/102167#102167 Answer by Bruce Westbury for Finite subgroups of Spin(9) Bruce Westbury 2012-07-13T18:58:35Z 2012-07-13T18:58:35Z <p>Although there exists an algorithm which will list all finite subgroups of $Spin(9)$ I suspect it is not effective. For background and references see</p> <p><a href="http://mathoverflow.net/questions/17072" rel="nofollow">http://mathoverflow.net/questions/17072</a></p> <p>I don't know if imposing the condition that the group acts freely on $S^{15}$ improves the situation. </p> http://mathoverflow.net/questions/102154/finite-subgroups-of-spin9/103871#103871 Answer by Maurizio Parton for Finite subgroups of Spin(9) Maurizio Parton 2012-08-03T15:00:35Z 2012-08-03T15:00:35Z <p>After some thinking, I came up with the following very partial answer, that I put here as a reference.</p> <p>Consider the diagonal inclusion of $\text{Sp}(1)$ in $\text{Spin}(9)$ given by sending a unitary quaternion $q\in\text{Sp}(1)$ into the map $L_q\in\text{Spin}(9)\subset\text{SO}(16)$, where $L_q$ is defined by $$L_q:\mathbb{H}^4\longrightarrow \mathbb{H}^4,\qquad L_q(h_1,h_2,h_3,h_4)=(qh_1,qh_2,qh_3,qh_4).$$</p> <p>If $q\neq 1$, any such $L_q$ acts without fixed points on $S^{15}$. Since the finite subgroups $G$ of $\text{Sp}(1)$ are known (polyhedrons classification), any such $G$ gives a finite subgroup $L_q(G)$ of $\text{Spin}(9)$ acting without fixed points.</p>