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Given a finite group $G$ acting on a vector space $V$ (in characteristic zero), is there a known algorithm to find a $G$-invariant basis for $V$? As a concrete example to work with, one of the systems I'm looking at is the dihedral group $D_6$ acting on $\mathbb{Q}^{10}$.

EDIT: All the representations I'm working with are going to be permutation representations.

edited Jul 8 2011 at 23:04

asked Jul 8 2011 at 22:18

Avi Steiner
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Do you mean, "I have a representation V which I know to be a permutation representation. How can I find a basis of V which is permuted by G?" – John Wiltshire-Gordon Jul 8 2011 at 22:34

@John: Yes. I guess I should add that to my question. – Avi Steiner Jul 8 2011 at 23:03

I think you need to tell us how much information about your action you have, and how much you want. For example, as you probably know, you can't answer this question by looking at the character alone. For example, ${\rm GL}(3,2)$ has two non-conjugate subgroups isomorphic to $S_4$, and the transitive permutation representations on these respective subgroups are inequivalent as permutation representations, but they have the same character. – Geoff Robinson Jul 9 2011 at 0:21

Bruce Westbury · Answer 1 · 2011-07-09 04:55:34Z UTC

As mentioned in the comments it is not entirely clear what information is available. If you have matrices for the representation and permutations for the permutation representation then you simply find an isomorphism of representations.

You find a homomorphism between two representations, $U$, $V$, by choosing a set of generators for the group and then solving $P.U(g)=V(g).P$, for each generator. These are linear equations in the entries of $P$.

Algorithm to get G-invariant basis

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