Endofunctor on the category V of finite-dimensional vector spaces
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 (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
over a field of characteristic zero.[1]
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 (i.e., a vector-valued polynomial in linear forms).
- Given linear maps
in
, the function
defined on
is a polynomial function with coefficients in
.
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 (to be precise, those of polynomial nature).
References