# Characters of a Group Action

1. Aug 11, 2011

### Kreizhn

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

Show that these characters have the following values on conjugacy classes of $S_4$

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

3. The attempt at a solution

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

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 $\lambda$ cycle, are we always ensured the number of fixed points of $S_\mu$ are the same? I think this is true (and must be for the table to make sense), and works because $S_\mu$ contains ALL the possible partitions of n by $\mu$.

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.