= Alternating-group restriction of Specht modules
An ordinary <Specht module> of non-self-conjugate shape stays <irreducible> on the <alternating group>, and conjugate shapes give the same restriction. A self-conjugate shape splits into two <irreducibles> of equal degree, interchanged by an odd <permutation>. The index-two inner-product identity is $\langle\chi^\lambda\downarrow,\chi^\mu\downarrow\rangle=\delta_{\lambda\mu}+\delta_{\lambda'\mu}$, and <Frobenius reciprocity> proves exhaustion. The statement applies for $n\ge2$; $A_1$ is trivial.
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