Some irreducible characters of the Symmetric group $S_n$

Question

I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in standard text of representation theory of symmetric groups. Where I can find these in Litrature? More precisely I am more intereseted in the irreducible modules corresponding to partitions $(n-4,2,2)$ and $(n-4,2,1,1)$. Thank you in advance.

Answer 1

You want to use character polynomials. Let $\mu$ be a partition of $k$. In your case, $k$ will be $4$. The character polynomial is a polynomial $$q_{\mu}(a_1, a_2, \ldots, a_k)$$ with the following property:

For any $n$ such that $n-k \geq \mu_1$, and any $\sigma \in S_n$, the character of $\sigma$ acting on the Specht module $\mathrm{Specht}(n-k, \mu_1, \mu_2, \ldots, \mu_r)$ is $q_{\mu}(a_1, \ldots, a_k)$, where $a_i$ is the number of $i$ cycles of $\sigma$.

The simplest example is $q_1 = a_1-1$. In other words, for any permutation $\sigma \in S_n$, the trace of $\sigma$ acting on $\mathrm{Specht}(n-1,1)$ is $\#(\mbox{$1$-cycles of $\sigma$})-1$.

What you want to know are $q_{22}$ and $q_{211}$. I don't know this theory well enough to compute them for you in a reasonable amount of time, but I would suggest looking at Garsia and Goupil and at Examples 1.7.13 and 1.7.14 in MacDonald's book.