Generic representation theory of finite fields in nondescribing characteristic

Abstract

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and charF is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all n≥0, of the categories of K[GLn(F)]-modules.As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if charK=0: the category Rep(F;K) will be semisimple.In the last section, we briefly discuss ‘q=1’ analogues and consider representations of various categories of finite sets.The main result follows from a 1992 result by L.G. Kovács about the semigroup ring K[Mn(F)].