Answer Happy

= Algebra Representation Theory Question. Please do not copy other solutions on this website. Consider the representation Q: S + GL(V) given by p(g)(:-ze;) = }=12g(1)@g(1) where V is a vector space of dimension n equipped with a basis ei.....en. The nontrivial proper Sn invariant subspace of V is the subspace generated by ejt...ten. Prove this representation decomposes as a direct sum of two irreducible representations W1 and W2 where W1 is the subspace genereated by ei + ez +e3. And the complementary subspace W2 of Vis S3-invariant and irreducible. (I believe W2 = span {{1,1,-2),(1,-1,0)) but am not sure how to find that and prove it is irreducible and S3-invariant)