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(p35, F&H) Which irreducible representation of $S_n$ remain irreducible when restricted to $A_n$? Which irreducible representation of $S_n$ are induced from $A_n$?

My sketch solution:

Restriction: We apply the bound: $$ \langle \chi_H , \chi_H \rangle \le |G:H| \langle \chi, \chi \rangle $$ with equality iff $\chi(g)$ vanishes for all $g \in G-H$.

So $\chi_{A_n}$ is irreducible iff $\chi \not= \chi \otimes sgn $ (since $S_n-A_n$ are precisely the odd elements).

Induction: By Frobenius Reciprocity, if $W$ is an irreducible $A_n$ representation, $$ \langle \chi_{Ind_{A_n}^{S_n} W} , \chi_{Ind_{A_n}^{S_n} W} \rangle _{S_n} = 1 + \langle \chi_{W}^\tau , \chi_W \rangle $$ where $\tau$ is some fixed non trivial transposition. We define $\chi^{\tau}_W(g):= \chi_W(\tau g \tau^{-1})$. Note that $\chi^{\tau}_W$ is also an irreducible $A_n$ character.

Thus, induced representation is irreducible if and only if $\chi^{\tau} \not= \chi$ for a fixed transposition.

The question is posed in an open sense. I am wondering if these two conclusion are correct & good enough - or we can even deduce more.

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