Symmetric Functions: Problem Set 7
  1. Let {uλ},{u′λ} (and similarly {vλ},{v′λ}) be a pair of dual bases of Λ with respect to the form ⟨,⟩. If ⃗u = ⃗vA, prove that ⃗v′ = ⃗u′A′.
  2. Let λ,μ ∈ Par(d). Prove that if λ is a refinement of μ, then λ occurs after μ in reverse lexicographic order. Is it true that μ dominates λ ?
  3. Prove Pieri’s rule (in the ring of symmetric functions in n variables, for large n): sλhk = ∑ μ∈λ⊗ksμ where λ ⊗ k is the set of partitions obtained by adding k boxes to the diagram of λ, no two in the same column. Deduce that the same equation holds in the ring Λ.
  4. Let A denote the space of alternating polynomials in n variables. Show that (i) The aγ for γ ∈ D(n) span A, (ii) multiplication by aδ defines an isomorphism of vector spaces Λn →A. Thus, A is a free Λn-module generated by aδ.
  5. Let δ = (n-1,n-2,⋅⋅⋅,0). Show that sδ = ∏ 1≤i<j≤n(xi+xj) in the ring Λn. Compute skδ for all k ≥ 1.
  6. Prove the identities:
    fλ = ∑ λ∈μ⊗1fμ (1)
    (1 + |λ|)fλ = ∑ μ∈λ⊗1fμ (2)
    where fλ is the number of standard Young tableaux of shape λ.