Symmetric Functions: Problem Set 11
  1. Let λ = (λ1,λ2,⋅⋅⋅) be a partition (written as an infinite list, with all but finitely many entries zero). Let m ≥ λ1,n ≥ λ1′. Show that the m + n numbers
    λ + n- i(1 ≤ i ≤ n ),n - 1 + j - λ′(1 ≤ j ≤ m ) i j

    are a permutation of {0,1,2,⋅⋅⋅,m + n - 1}.

  2. Let λ,μ be partitions of n such that λ covers μ in the dominance order, i.e., λ > μ and if ν is such that λ ≥ ν ≥ μ, then ν = λ or ν = μ. Show that λ can be obtained by removing one box from the jth row of μ and moving it to the ith row, for some i < j.
  3. A matrix of non-negative real numbers is said to be doubly stochastic if its row and column sums are all equal to 1. Let λ,μ be partitions of n. Show that λ dominates μ if and only if there exists a doubly stochastic n×n matrix M such that Mλ = μ (where λ,μ are regarded as column vectors of length n).
  4. Let λ be a partition. The hook-length of λ at x = (i,j) ∈ λ is defined to be
    h(x) = (λi - i) + (λ′j - j)+ 1.

    The content of x is defined to be c(x) = j - i. Prove that

    ∑ (h (x )2 - c(x)2) = |λ|2. x∈λ

  5. Show that the set of partitions of n under the dominance order is a lattice. In other words, each pair of partitions of n has a greatest lower bound and a least upper bound.
  6. Let m ≥ 1.
    1. Show that the set D(m) of strictly decreasing m-tuples of nonnegative integers is in bijection with the set of partitions with at most m parts under the map λ↦→λ† where λi† = λi - (m - i) for all i.
    2. Let λ ∈D(m), 1 ≤ i ≤ m and p ≥ 1. Define uj = λj for j≠i, and ui = λi - p. Assume that the {uj : 1 ≤ j ≤ m} are all distinct, non-negative integers. Let μ ∈D(m) denote the tuple obtained by rearranging the uj in descending order. Describe the image μ† of μ under the above bijection.
  7. Let P(n) denote the set of partitions of n, and ℕ the set of positive integers. For each r ≥ 1, let
    a(r,n) = #{(λ,i) ∈P(n) × ℕ : λi = r}
    b(r,n) = #{(λ,i) ∈P(n) × ℕ : mi(λ) ≥ r}
    Show that
    a (r,n) = b(r,n) = p(n - r)+ p(n - 2r)+ ...

    where p(m) is the number of partitions of m. Deduce that

    ∏ ∏ imi(λ) = ∏ ∏ m (λ )! i≥1 i≥1 i λ∈P(n) λ∈P (n)

  8. Keep the above notation. Let h(r,n) = #{(λ,x) : λ ∈P(n),x ∈ λ and h(x) = r}, where h(x) is the hook-length of λ at x. Show that h(r,n) = ra(r,n).