Trace of a permuted tensor power ($\mathrm{tr}(\sigma {\xi }^{\otimes n})={\prod }_r\mathrm{tr}({\xi }^r{)}^{{\alpha }_r}$)

If $\sigma$ has ${\alpha }_r$ cycles of length $r$, contraction of matrix entries around its cycles gives $\mathrm{tr}(\sigma {\xi }^{\otimes n})={\prod }_r\mathrm{tr}({\xi }^r{)}^{{\alpha }_r}$. 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.