Splitting of composite functors. (Q2386061)

The framework is the category of functors between vector spaces over a prime field. The symmetric power \(S^n\) is the functor obtained as the quotient of the tensor power by the action of the symmetric group (under permutation of factors). A functor \(B\) is said to be \(p\)-Boolean if it takes values in an algebra in which \(x^p= x\). The notion of polynomial and analytic functors leads to the Taylor filtration of a functor. The main result shows that the Taylor filtration of \(S^n\) splits after (right) composition with a \(p\)-Boolean functor \(S^n\circ B\). As applications of such splittings are computations of extensions appearing in the study of the cohomology of Eilenberg-MacLane spaces.