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

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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

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If “finite vector spaces” is replaced by “finite sets”, one gets the notion of combinatorial species (to be precise, those of polynomial nature).

References

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  1. ^ Macdonald 1995, Ch. I, Appendix A: 5.4.
  • Macdonald, Ian G. (1995). Symmetric functions and Hall polynomials. Oxford: Clarendon Press. ISBN 0-19-853489-2. OCLC 30733523.MR1354144