= Central average of representation matrices at squares
{title2=$\sum_{g\in G}D(g^2)=\frac{|G|\nu(D)}{\dim D}I$}
For a complex <irreducible representation> $D$ of a <finite group> $G$, the sum $A=\sum_gD(g^2)$ commutes with every $D(h)$: conjugation sends its summands to $D((hgh^{-1})^2)$, merely permuting the summation index. <Schur's lemma> gives $A=cI$. Taking the <trace> yields $c\dim D=\sum_g\chi(g^2)=|G|\nu(D)$, proving the identity. Here $\nu(D)$ is the <Frobenius-Schur indicator>. For a reducible representation the sum is still central in its image, but need not be one scalar on all irreducible constituents.
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