2.7 The character table of S₅

Let $G=S_5$. We have the trivial representation $\mathrm{𝟙}$, the sign representation $ϵ$, and the permutation representation $V\cong \mathrm{𝟙}\oplus W$, and its twist, as before. So we can start off the character table:

We then try ${\mathrm{\Lambda }}^{2}W$, which has character as shown (sadly, this is equal to its twist by $ϵ$). This is an irreducible character.

We can also try ${\mathrm{Sym}}^{2}W$, which has character below; it isn’t irreducible.

By taking inner products with the characters we’ve already found, we see that

$${\mathrm{Sym}}^2\chi =\mathrm{𝟙}\oplus \chi \oplus \psi$$

where $\psi$ is an irreducible character. We get one more from twisting $\psi$.

This gives all of the irreducible characters, which we assemble into Table 2.