= Trace of a permuted tensor power
{title2=$\operatorname{tr}(\sigma\xi^{\otimes n})=\prod_r\operatorname{tr}(\xi^r)^{\alpha_r}$}
If $\sigma$ has $\alpha_r$ cycles of length $r$, contraction of matrix entries around its cycles gives $\operatorname{tr}(\sigma\xi^{\otimes n})=\prod_r\operatorname{tr}(\xi^r)^{\alpha_r}$. The formula holds for every <endomorphism>, not only diagonalizable ones. The <Schur–Weyl duality> decomposition equates it with a sum of products of Specht and Schur <characters>.
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