Generic representation theory of finite fields in nondescribing characteristic

Nicholas J. Kuhn

Published 2014-05-01, updated 2014-05-07Version 2

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 Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,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 Char K = 0: the category Rep(F;K) will be semisimple. In a 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.Kovacs about the semigroup ring K[M_n(\F)].