Let $X$ be a complex manifold, let $(E,h)$ be a Hermitian holomorphic vector bundle of rank $r$ on $X$, and let $m\ge 1$ be an integer. Let $E^{\otimes m}$ be the $m$-fold tensor power bundle with [tensor product](/page/Tensor%20Product) metric $h^{\otimes m}$, and let $S^mE\subset E^{\otimes m}$ be the symmetric power subbundle with the induced Hermitian metric $h_{S^mE}$. For each $x\in X$, let $\rho_{m,x}:\operatorname{End}(E_x)\to \operatorname{End}(S^mE_x)$ be the infinitesimal symmetric power representation defined by applying an endomorphism to one tensor factor at a time and summing over the $m$ tensor positions on decomposable symmetric tensors. Equivalently, these fibrewise maps assemble to a bundle map $\rho_m:\operatorname{End}(E)\to \operatorname{End}(S^mE)$. Then the Chern curvature of $(S^mE,h_{S^mE})$ is given by

\begin{align*} \Theta(S^mE,h_{S^mE})=\rho_m\bigl(\Theta(E,h)\bigr). \end{align*}

Equivalently, for every $x\in X$, every $\xi\in T_xX$, and every $u\in S^mE_x$,

\begin{align*} \bigl(i\Theta(S^mE,h_{S^mE})(\xi,\overline{\xi})u,u\bigr)_{h_{S^mE}}=\bigl(i\rho_{m,x}(\Theta(E,h)(\xi,\overline{\xi}))u,u\bigr)_{h_{S^mE}}. \end{align*}

In particular, if $(E,h)$ is Griffiths semipositive, then $(S^mE,h_{S^mE})$ is Griffiths semipositive.