Symmetric Functions: Problem Set 9
  1. Show that the definitions of the induced representation IndHG(W) (given in lecture) make it a representation of G.
  2. Prove Frobenius reciprocity; given a H-map f : W → U, there is a unique G-map f˜ such that ˜f∘ i = f where i is the natural inclusion from W → IndHG(W).
  3. For each irrep W of S3, compute the decomposition of IndS3S4W into irreps of S4, using the inner product version of Frobenius reciprocity.
  4. If X is a transitive G-set and H is the stabilizer of an element of X, prove that IndHG(W) is isomorphic to ℂ[X], where W is the one dimensional trivial representation of H.
  5. Let W be an irrep of H. Prove that ⟨IndHG(W),IndHG(W)⟩≤ [G : H].