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Characters of a Group Action

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider the symmetric group on n-letters denoted [itex] S_n [/itex]. For [itex] \lambda [/itex] a positive integer partition of n (under the usual cycle type notation) define [itex] \sigma_\lambda[/itex] to be the character of the permutation representation of [itex] S_n [/itex] acting on the set of all ways to divide n into sets of size [itex] \lambda[/itex].

    Show that these characters have the following values on conjugacy classes of [itex] S_4 [/itex]

    [tex] \begin{array}{l|rrrrr}
    & [1 1 1 1] & [2 1 1] & [2 2] & [3 1] & [4] \\ \hline
    \sigma_4 & 1 & 1 & 1 & 1 & 1 \\
    \sigma_{3,1} & 4 & 2 & 0 & 1 & 0 \\
    \sigma_{2,2} & 6 & 2 & 2 & 0 & 0 \\
    \sigma_{2,1,1} & 12 & 2 & 0 & 0 & 0 \\
    \sigma_{1,1,1,1} & 24 & 0 & 0 & 0 & 0
    \end{array}
    [/tex]

    3. The attempt at a solution

    Even just trying to figure out what "the character of the permutation representation of [itex] S_n [/itex] acting on the set of all ways to divide n into sets of size [itex] \lambda[/itex]" means is hurting my head, though I think I might get it. In particular, let [itex] \mu [/itex] be another partition of n and take [itex] S_\mu [/itex] to be the set of all [itex] \mu [/itex] partitions of [itex] \{ 1, 2, \cdots, n \} [/itex]. Then [itex] (\lambda,\mu) [/itex] element of the table is the number of fixed points of [itex] S_\mu [/itex] under permutations of cycle type [itex] \lambda [/itex].

    First Question: Have I interpreted this correctly? Some scratch calculations imply the numbers in the table may actually be the number of orbits (for any fixed element) of the action.

    Second Question: Is this well defined? Namely, by choosing an arbitrary [itex] \lambda [/itex] cycle, are we always ensured the number of fixed points of [itex] S_\mu [/itex] are the same? I think this is true (and must be for the table to make sense), and works because [itex] S_\mu [/itex] contains ALL the possible partitions of n by [itex] \mu[/itex].

    Third Question: How does one calculate these table values in a reasonable way. I can kind of see it, but it's not precise in my head.
     
  2. jcsd
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