Basic Symmetric Group Representation Question

zer0skill

If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the representation. Also, the subspace in which all coordinates sum to zero is invariant. But are there any others that are independent of these ones?

conquest

The irreducible representations of the symmetric group in n letters are actually quite well understood and to find them all actually for a given vector space over C. They are even paramtrized by partitions of n. I couldn't give the answer explicitly very quickly, but just look for books on it keeping in mind the terms Specht module and Young diagram.

morphism

No, there are no others: the two subspaces you mention are irreducible, have dimensions 1 and n-1 (resp.), and intersect in zero.