A stratification of generic representation theory and generalized Schur algebras, by Nicholas J. Kuhn

Let F(q) be the category whose objects are functors from finite dimensional Fq-vector spaces to Fq-vector spaces, and with morphisms the natural transformations between such functors. (Fq is the finite field of order q and characteristic p.)

We define an infinite lattice of thick subcategories of F(q). Our main result then identifies various subquotients as categories of modules over products of symmetric groups, via recollement diagrams.

Our lattice of thick subcategories is a refinement of the Eilenberg-MacLane polynomial degree filtration of F(q) which has been extensively used in the algebraic K-theory literature. Our main theorem refines and extends earlier results of Pirashvili.

If q > r-1, one of the subcategories is the Friedlander-Suslin category P^r of `strict polynomial functors of degree r', equivalent to the category of modules over the Schur algebra S(n,r) with n > r-1. Our results can thus also be viewed as refining and extending the classic relationship between S(n,r)-modules and Sigma_r-modules via the Schur functor. In fact, (essentially) all our subcategories are equivalent to categories of modules over various finite dimensional algebras, and our lattice can be interpreted in terms of lattices of idempotent two sided ideals in these generalized Schur algebras.

Applications include a simple proof, free of algebraic group theory, of a Steinberg Tensor Product Theorem. Our tensor product theorem is then used to study when various of our generalized Schur algebras are Morita equivalent.

We use two technical tricks which may be of some independent interest. Firstly, we `twist' by an action of the Galois group Gal(Fq;Fp) to be able to work entirely with vector spaces over the prime field Fp, making discussions of `base change' unnecessary. Secondly, we systematically use lax symmetric monoidal functors to define our subcategories.

Nicholas J. Kuhn <njk4x@virginia.edu>