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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.

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  • $\begingroup$ you heard about orthogonality relations? have you done character table for $S_4$? you know some thing about induced representations? $\endgroup$ – user87543 Feb 16 '14 at 5:01
  • $\begingroup$ @PraphullaKoushik I know about orthogonality relations and I have done character table of $S_3$, You mean some irreps of $S_4$ induce irreps for $S_5$ ? $\endgroup$ – the8thone Feb 16 '14 at 5:40
  • $\begingroup$ Yes.. I mean some irreps of $S_4$ induce irreps of $S_5$ $\endgroup$ – user87543 Feb 16 '14 at 6:51
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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.

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