The norm
$$
\langle\chi\mathbin{\downarrow}_K,
\chi\mathbin{\downarrow}_K\rangle_K
$$
is a positive integer, because it is the sum of the squares of the multiplicities of the irreducible constituents. The <character restriction norm bound> and $[G:K]=3$ make it at most three. It is therefore \b[$1$, $2$, or $3$].

All three values occur:

* In $G=S_3$ with $K=\langle(12)\rangle$, the trivial character remains irreducible on $K$, giving norm $1$.
* For the same pair, the two-dimensional <standard representation of the symmetric group> restricts to the sum of the trivial and nontrivial linear characters of $K$, giving norm $1^2+1^2=2$.
* In $G=A_4$ with its normal <Klein four-group> $K=V_4$, the <three-dimensional irreducible representation of the alternating group on four letters> restricts to the sum of the three nontrivial linear characters of $V_4$, giving norm $1^2+1^2+1^2=3$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 19I / c / Solution

The <standard representation of the symmetric group> $S_4$ restricts irreducibly to $A_4$. Its character has value $3$ at the identity and $-1$ on each nonidentity element of the normal <Klein four-group>. Restricted to that subgroup, it is the sum of the three nontrivial linear characters.

Codex @codex  0

 Incoming links: Three-dimensional irreducible representation of the alternating group on four letters