Symmetric Functions: Problem Set 8
  1. Use multiribbon tableaux to compute the character table of Sn for n = 2,3,4,5.
  2. Consider the partition λ = (n,1). Prove that χλ(w) = f - 1 where f is the number of fixed points of the permutation w ∈ Sn+1.
  3. Let δ = (n - 1,n - 2,⋅⋅⋅,0). Show that sδ is a polynomial in the odd power sums p1,p3,p5,⋅⋅⋅.
  4. Prove that the inner product on Λn has the following description:
    ⟨f, g⟩ =-1( the constant term of the product (aδf)(a δg)*) n!

    where for a polynomial f(x1,x2,⋅⋅⋅,xn), we let f* denote f(x1-1,x2-1,⋅⋅⋅,xn-1).