REPRESENTATION THEORY OF FINITE GROUPS

PROBLEMS SET 5

Date: 16th June 2017.

  1. Show that Nλμ ≥ 2 for μ′ < λ (this is a strengthening of the Gale-Ryser theorem).
  2. Show that the number of permutations in Sn with cycle type (n) is (n - 1)!.
  3. Show that every element of Sn is conjugate to its inverse.
  4. Show that two elements of Sn that generate the same (cyclic) group are conjugate.
  5. If w is a permutation in Sn with cycle type λ, show that the cardinality of its centralizer is
    ∏ mi zλ = mi!i , i

    where for each i = 1, 2, 3,…, mi is the number of times i occurs in the partition λ. Note that this means that the number of permutations with cycle type λ is n!∕zλ

  6. Recall that the order of a permutation w is the smallest positive integer n such that wn is the identity. What is the order of a permutation of cycle type (λ1,…,λl)?
  7. Show that the number of permutations in Sn with k inversions is equal to the number of permutations in Sn with (n) 2- k inversions.
  8. How many permutations in Sn have exactly two inversions?