Symmetric Functions: Problem Set 31
  1. Prove that the map ω : ei↦→hi defined in lecture is indeed an involution (we only did this by example).
  2. Define fλ := ω(mλ). The fλ are called the forgotten symmetric functions. Show that they form a basis of Λ. Compute fλ for |λ| = 1,2,3.
  3. Compute the change of basis matrix when eλ is written in terms of the fμ.
  4. Let xi = 1 for 1 ≤ i ≤ n and xi = 0 for i > n. Show that under this substitution, er = ( ) nr. Find the values of hr and mλ for r ≥ 1 and λ ∈ Par.
  5. Let us substitute xi = 1∕n for 1 ≤ i ≤ n and xi = 0 otherwise. Now take the limit as n →∞. Compute the values of er,hr and mλ in this limit.
  6. Prove that:
    en = det(h ) 1- i+j1≤i,j≤n
    hn = det(e1-i+j)1≤i,j≤n
  7. If hn = n for each n ≥ 1 (recall that the hn are algebraically independent generators, and one can assign any value to them), the sequence (en)n≥1, is periodic with period 3.