Polynomial functor

In algebra, a polynomial functor is an endofunctor on the category ${\displaystyle {\mathcal {V}}}$ of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers ${\displaystyle V\mapsto \operatorname {Sym} ^{n}(V)}$ and the exterior powers ${\displaystyle V\mapsto \wedge ^{n}(V)}$ are polynomial functors from ${\displaystyle {\mathcal {V}}}$ to ${\displaystyle {\mathcal {V}}}$; these two are also Schur functors.

The notion appears in representation theory as well as category theory (the calculus of functors). In particular, the category of homogeneous polynomial functors of degree n is equivalent to the category of finite-dimensional representations of the symmetric group ${\displaystyle S_{n}}$ over a field of characteristic zero.^[1]