REPRESENTATION THEORY OF FINITE GROUPS

PROBLEMS SET 4

Date: 15th June 2017.

  1. Find the 5 × 4 matrix A to which the VRSK algorithm would associate the SSYTs:
    P = 1 1 2 3and Q = 1 1 3 4. 2 4 2 5

  2. Find the matrices A for which both P and Q have the same shapes as their types.
  3. If λ and μ are partitions such that μ ≤ λ (in the reverse dominance order), then show that μ comes after λ in lexicographic (dictionary) order.
  4. List all the partitions of n in reverse lexicographic order:
    λ(1),...,λ(p),

    where p is the number of partitions of n. Define p × p matrices

    M = (M λ(i),λ(j)), and K = (Kλ(i)λ(j)).

    Compute M and K for n = 2, 3, 4. Verify that M = K′K.

  5. Recall that the conjugate of a partition λ is defined by
    λ′ = # {j | λ ≥ i}. i j

    Show that conjugation reverses dominance:

    λ ≤ μ if and only if μ′ ≤ λ′.

  6. Using the notation of problem (4), given n, define a p × p matrix:
    N = (N λ(i),λ(j)), and J = (δλ(i),λ(j)′).

    Here δ is the Kronecker delta symbol. For n = 2, 3, 4, verify that N = K′JK.