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 power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n.
More precisely, the notion exists at least in the ...
s
and the
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
s
are polynomial functors from
to
; these two are also
Schur functors.
The notion appears in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
as well as
category theory (the
calculus of functors In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes the sheafification of a presheaf. This sequ ...
). In particular, the category of homogeneous polynomial functors of degree ''n'' is equivalent to the
category of finite-dimensional representations of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
over a field of characteristic zero.
Definition
Let ''k'' be a
field of
characteristic zero and
the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of finite-dimensional ''k''-
vector spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and ''k''-
linear maps
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. 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 mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
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
*
Functors
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