How can we compute restrictions from a character table?

Question

I would like to how to, when given a character table, calculate the restriction.

$Res_H^G : Rep(G) \rightarrow Rep(H)$.

For example:

Let $G=S_4$ whose character table is given below (see picture), and $H=<(123)>$, the cyclic group generated by the cycle $(123)$. Compute the restrictions $Res_H^G\chi_i$ for $i=1, ..., 5$

Can these be computed from character; I assume so since we are given the character table?

Answer 1

Yes, they can. As Tobias wrote in a comment, they follow directly by restricting the table to the columns $1$ and $(123)$ that represent the conjugacy classes that make up the normal subgroup. Denoting the three irreducible characters of $H$ by $X_k=\exp(2k\pi \mathrm i/3)$, you can use the restricted characters to find how the irreducible characters of $G$ decompose into irreducible characters of $H$:

\begin{align} \operatorname{Res}^G_H\xi_1&=X_1\;,\\ \operatorname{Res}^G_H\xi_2&=X_1\;,\\ \operatorname{Res}^G_H\xi_3&=X_1+X_2+X_3\;,\\ \operatorname{Res}^G_H\xi_4&=X_1+X_2+X_3\;,\\ \operatorname{Res}^G_H\xi_5&=X_2+X_3\;. \end{align}