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:- For every pair of vector spaces X, Y in, the map is a polynomial mapping.
- Given linear maps in, the function defined on is a polynomial function with coefficients in.