Polynomial functor explained

In algebra, a polynomial functor is an endofunctor on the category

l{V} of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers

V \mapsto \operatorname{Sym} ^{n(V)} and the exterior powers V \mapsto \wedge ^{n(V)} are polynomial functors from l{V} to l{V} ; these two are also Schur functors.S _{n} over a field of characteristic zero.

Definition

Let k be a field of characteristic zero and

l{V} the category of finite-dimensional k-vector spaces and k-linear maps. Then an endofunctor F\colon l{V} \to l{V} is a polynomial functor if the following equivalent conditions hold:

• For every pair of vector spaces X, Y in

l{V} , the map F\colon \operatorname{Hom}(X, Y) \to \operatorname{Hom}(F(X), F(Y)) is a polynomial mapping (i.e., a vector-valued polynomial in linear forms).

• Given linear maps

f _{i:} X \to Y, 1 \le i \le r in l{V} , the function ( λ _{1,} ... , λ _{r)} \mapsto F( λ _{1} f _{1} + … + λ _{r} f _{r)} defined on k ^{r} is a polynomial function with coefficients in \operatorname{Hom}(F(X), F(Y)) .

A polynomial functor is said to be homogeneous of degree n if for any linear maps

f _{1,} ... , f _{r} in l{V} with common domain and codomain, the vector-valued polynomial F( λ _{1} f _{1} + … + λ _{r} f _{r)} 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

• Book: Macdonald, Ian G. . Ian G. Macdonald . Symmetric functions and Hall polynomials . Clarendon Press . Oxford . 1995 . 0-19-853489-2 . 30733523 .