Symmetric Functions: Problem Set 4
  1. Let p0 = 0 and pn = ∑ ixin for n ≥ 1. Define
    ∑ n P (t) = pnt n≥0

    Show the following:

    P(t) = ∑ i txi 1---tx- i = t d dt∑ i log(1 - txi)-1 (1)
    P(t) = t ′ H-(t) H (t) (2)
    P(-t) = -tE′(t) E-(t)- (3)
    ∑ r=1np rhn-r = nhn(n ≥ 1) (4)
    ∑ r=1n(-1)r-1p ren-r = nhn(n ≥ 1) (5)
  2. Prove that ω(pr) = (-1)r-1pr and that ω(pλ) = ϵλpλ where ϵλ is the sign of wλ, an element of the symmetric group of cycle type λ.
  3. The trace of the involution ω on Λd is the number of even partitions of d minus the number of odd partitions of d (a partition λ is even if ϵλ = 1 and odd otherwise).
  4. Let the graded trace of ω be:
    ∑ ζ(q) = Tr (ω| d)qd d≥0 Λ

    Prove that ζ(q) = 1 ------------2------3------4---- (1 - q)(1 + q )(1 - q )(1+ q )⋅⋅⋅.

  5. Using the fact that
    1 -----------3------5-----= (1+ q)(1+ q2)(1 + q3)⋅⋅⋅ (1 - q)(1- q )(1- q ) ⋅⋅⋅

    prove that Tr(ω|Λd) equals the number of self conjugate partitions of d.