character table S_9

Question

I'm trying to find a character table for a symmetric group S_9. I found trivial, alternating, permutation and characters derived by the multiplication permutation character with the alternating character. Also, I found character which we could find with the formulas on the photo below, and that we generate them using their multiplication with alternating character. S_9 have 30 irreducible character. Is there any suggestion how to find the rest of them? I know that somehow the subgroups of S_9 can help in finding the rest of, but I do not know which is the best one to do it. Thank you.

Answer 1

The characters of the symmetric group $S_n$ were discovered by Frobenius. His approach was to take characters generated in a certain way, and prove that integer linear combinations of them contained all the characters of $S_n$.

What we did was to consider subgroups isomorphic to $S_{n_1}\times\cdots\times S_{n_k}$ of $S_n$ where $n_1+\cdots+n_k=n$, and induce the trivial character on each these subgroups to the whole group. That gave sufficient characters on $S_n$ to derive all of them.

Of course Frobenius did all this in a systematic way, but with patience, you could do it an an ad hoc way when $n=9$. There are only thirty characters to find, of course.