I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily?
The best approach I have so far is as follows:
Find a set of generators of $S_n$, eg. $(12)$ and $(123...n)$ and find the common invariant subspaces of their corresponding matrices. To find the common invariant subspaces of two matrices, find their individual eigendecompositions and 'merge' them.
This approach is not very good since computing the eigendecomposition is messy (it can't easily be done exactly with a computer) and should also not be necessary since I know all irreducible representations are rational.
I'm interested in either a better approach or a way to make the current approach useful. I really do want the coordinates of the invariant subspaces out, and as input I have the matrices for all group elements.