Polynomial functor

In algebra, a polynomial functor is a functor on the category ${\mathcal {V}}$ of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric and exterior powers $V\to \operatorname {Sym} ^{n}(V),V\mapsto \wedge ^{n}(V):{\mathcal {V}}\to {\mathcal {V}}$ are polynomial functors.

Definition

Let k be a field of characteristic zero. Then an endofunctor $F:{\mathcal {V}}\to {\mathcal {V}}$ is a polynomial functor if the following equivalent conditions hold:

• For every vector spaces X, Y in ${\mathcal {V}}$, the map $F:\operatorname {Hom} (X,Y)\to \operatorname {Hom} (F(X),F(Y))$ is a polynomial mapping (i.e., a vector-valued polynomial in linear forms).
• Given linear maps $f_{i}:X\to Y,\,1\leq i\leq r$ in ${\mathcal {V}}$, the function $(\lambda _{1},\dots ,\lambda _{r})\mapsto F(\lambda _{1}f_{1}+\dots +\lambda _{r}f_{r})$ defined on $k^{n}$ is a polynomial function with coefficients in $\operatorname {Hom} (F(X),F(Y)$.

References

• Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MR 1354144