# Computing centralizers in Lie groups

Let $G$ be a real semisimple Lie group. Really, I only care about $\text{SL}(n,\mathbb{R})$ and $\text{Sp}(2n,\mathbb{R})$.

I'd like to perform a computer search for a finite group with a certain property (that property is not important here). To do this, I need to be able to efficiently compute the isomorphism classes (as Lie groups) of the centralizers of certain finite subgroups of $G$. The finite groups involved might be rather large.

This brings me to two related questions. First, where can I find algorithms for this? Second, what would be the best form in which to generate my finite subgroups? For instance, is it fine to just give generators, should I give a list of all the matrices in the finite group, do I also need a presentation of the finite groups, etc?

-
1. Such a finite group comes with an $n$-dim or $2n$-dim complex representation, perhaps reducible, and the centralizer inside that $GL(n$ resp. $2n)$ just comes from Schur's lemma and the multiplicities of the irreps. Then it's trickier, I suppose, to see how those centralizers intersect your $G$. –  Allen Knutson Jun 11 '11 at 1:07
2. Generators should be enough -- centralizing the finite group is the same as commuting with its generators. The centralizer of a single element of finite order is easy to compute from Borel-de Siebenthal theory, but intersecting them seems likely to be hard, a priori. –  Allen Knutson Jun 11 '11 at 1:09