Polynomial functor


In algebra, a polynomial functor is an endofunctor on the category of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers and the exterior powers are polynomial functors from to ; these two are also Schur functors.
The notion appears in representation theory as well as category theory. In particular, the category of homogeneous polynomial functors of degree n is equivalent to the category of finite-dimensional representations of the symmetric group over a field of characteristic zero.

Definition

Let k be a field of characteristic zero and the category of finite-dimensional k-vector spaces and k-linear maps. Then an endofunctor is a polynomial functor if the following equivalent conditions hold:
A polynomial functor is said to be homogeneous of degree n if for any linear maps in with common domain and codomain, the vector-valued polynomial is homogeneous of degree n.

Variants

If “finite vector spaces” is replaced by “finite sets”, one gets the notion of combinatorial species.