Symmetric power bundle

Let $\pi:E\to M$ be a smooth vector bundle of rank $r$ over a smooth manifold. For an integer $k\ge 0$, the k-th symmetric power bundle of $E$ is the vector bundle

defined fiberwise by

the $k$-th symmetric power of the vector space $E_x$ (i.e. the quotient of $E_x^{\otimes k}$ by the action of the symmetric group permuting factors).

In local frames, if $(e_1,\dots,e_r)$ is a local frame of $E|_U$, then the symmetrized tensors built from the $e_i$ give a local frame of $(S^kE)|_U$. Under a change of frame with transition matrix $g$, the induced transition on $S^kE$ is the symmetric power representation $S^k g$. This can be constructed systematically from the local description of the tensor product bundle $E^{\otimes k}$.

Functoriality holds: a bundle map $\Phi:E\to F$ over $\mathrm{id}_M$ induces $S^k\Phi:S^kE\to S^kF$ fiberwise.