Symmetric Functions: Problem Set 2
  1. Expand the following product and write it as a power series in t with coefficients being symmetric functions in x:
    ∏∞ (1+ txi) i=1

  2. Expand the following product, writing it as a power series in t, with coefficients being products of symmetric functions of x and y:
    ∏∞ (1 + txiyj) i,j=1

  3. Let en(x) denote the monomial symmetric function m(1,1,⋅⋅⋅,1)(x). For a partition λ = (λ1,λ2,⋅⋅⋅,λp) of d, define:
    ∏p eλ(x) = eλi(x). i=1

    Write eλ(x) as a linear combination of the basis {mμ(x) : μ ∈ Par(d)} for d = 1,2,3,4 and each choice of λ.

  4. Expand the following product and write it as a power series in t with coefficients being symmetric functions in x:
    ∏∞ (1 - txi)- 1 i=1

  5. Expand the following product, writing it as a power series in t, with coefficients being products of symmetric functions of x and y:
    ∏∞ (1 - txiyj)-1 i,j=1

  6. Let hn(x) := ∑ μ∈Par(n)mμ(x) denote the complete homogeneous symmetric function. For a partition λ = (λ1,λ2,⋅⋅⋅,λp) of d, define:
    ∏p hλ(x) = hλi(x). i=1

    Write hλ(x) as a linear combination of the basis {mμ(x) : μ ∈ Par(d)} for d = 1,2,3,4 and each choice of λ.

  7. Let n ≥ 1; recall that we have a map ρ : Λd → Λnd obtained by setting all variables xi,i > n equal to zero. If {bi} is a basis of Λd, show that it is not true that the nonzero elements in {ρ(bi)} must be a basis of Λnd ? Can you give an example of a basis of Λd for which this does hold ?
  8. Express the product m(2,1)m(1,1) as a linear combination of monomial symmetric functions.
  9. For positive integers p,q show that
    min(p,q)( ) ∑ p + q - 2k m (1p)m (1q) = p- k m (2k,1p+q-2k) k=0

  10. Prove that mλ can be written as a polynomial in the e1,e2,⋅⋅⋅, i.e, there is a polynomial fλ ∈ k[u1,u2,⋅⋅⋅] such that mλ = f(e1,e2,⋅⋅⋅).