Polynomial functor

In algebra, a polynomial functor is an <a href="/facts/Endofunctor/l2u3rIBE">endofunctor</a> on the <a href="/facts/Category_of_vector_spaces/AnnF9JKf">category 
 
 
 
 
 
 V
 
 
 
 
 {\displaystyle {\mathcal {V}}}
 
 of finite-dimensional vector spaces</a> that depends polynomially on vector spaces. For example, the <a href="/facts/Symmetric_power/muvAogoJ">symmetric powers</a> 
 
 
 
 V
 ↦
 
 Sym
 
 n
 
 
 ⁡
 (
 V
 )
 
 
 {\displaystyle V\mapsto \operatorname {Sym} ^{n}(V)}
 
 and the <a href="/facts/Exterior_power/rdN4TvpR">exterior powers</a> 
 
 
 
 V
 ↦
 
 ∧
 
 n
 
 
 (
 V
 )
 
 
 {\displaystyle V\mapsto \wedge ^{n}(V)}
 
 are polynomial functors from 
 
 
 
 
 
 V
 
 
 
 
 {\displaystyle {\mathcal {V}}}
 
 to 
 
 
 
 
 
 V
 
 
 
 
 {\displaystyle {\mathcal {V}}}
 
; these two are also <a href="/facts/Schur_functor/oeNeBBkI">Schur functors</a>.
The notion appears in <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> as well as <a href="/facts/Category_theory/6zS3DXPa">category theory</a> (the <a href="/facts/Calculus_of_functors/CNBfsQFo">calculus of functors</a>). In particular, the category of homogeneous polynomial functors of degree n is equivalent to the <a href="/facts/Category_of_representations/Lxr3DLnt">category of finite-dimensional representations</a> of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
 over a field of characteristic zero.

Polynomial functor open-in-new

Polynomial functor