This might be a very elementary question in representation theory, but I dare to ask
Suppose I am asked to complete the character table of $S_5$, I know it has 7 conjugacy classes as follows :
$\Gamma_1=\{\phi\}, \ \ \Gamma_2=\{(12)\}, \ \ \Gamma_3=\{(123)\}, \ \ \Gamma_4=\{(1234)\}, \ \ \Gamma_5=\{(12345)\}, \ \ \Gamma_6=\{(12)(34)\}, \ \ \Gamma_7=\{(12)(345)\}$
Where $\Gamma_1$ has $1$ element,$\Gamma_2$ has $10$ elements,$\Gamma_3$ has $20$ elements,$\Gamma_4$ has $30$ elements,$\Gamma_5$ has $24$ elements,$\Gamma_6$ has $15$ elements and $\Gamma_7$ has $20$ elements, which add up to $5!=120$.
Therefore I have to find $7$ irreducible representations for $S_5$. The first two are not difficult to see , the "Trivial" and the "sign" representations which are both one-dimensional.
Now my question is at this point, how one tries to find the other five irreps ? is it a matter of wisdom and experience ? or there is some concrete pattern/relation that I have to find ?
How does one make sure that how many one dimensional irreps are to be found ? what about irreps of higher dimensions ?
I would be very thankful if somebody can help me understand these things.
Best Answer
The conjugacy classes you have provided correspond to partitions of $5$ by taking cycle types. The partitions of $5$ naturally index the irreducible representations of $S_5$. The irreducible representation of $S_5$ corresponding to a partition $\lambda$ is called the Specht module $S^{\lambda}$. These representations can be described explicitly. A reference is "Young Tableaux" by Fulton.
The characters of Specht modules are reasonably computable, at least for $S_5$. There is the Jacobi-Trudi formula expressing the Schur polynomial $s_{\lambda}$ in terms of the polynomials $h_{\mu}$. Using the isomorphism between the representation ring $R=\bigoplus_{n=1}^{\infty}R_n$ ($R_n$ being the representation ring of $S_n$) and the ring $\Lambda$ of symmetric functions, you obtain a corresponding formula for the character of $S^{\lambda}$.
For your other question, a one-dimensional representation $\varphi:G\rightarrow\mathbb{C}^*$ of a finite group $G$ always factors through the abelianization $G/[G,G]$ of $G$. (Here, $[G,G]$ is the commutator subgroup of $G$.) So, you can reduce to the case of finding all one-dimensional representations $\psi:G/[G,G]\rightarrow\mathbb{C}^*$. In your case, the commutator subgroup of $S_5$ is $A_5$, so that the abelianization is isomorphic to $\mathbb{Z}_2$. The latter group has exactly two one-dimensional representations up to isomorphism.