Elsevier

Advances in Mathematics

Volume 272, 26 February 2015, Pages 598-610
Advances in Mathematics

Generic representation theory of finite fields in nondescribing characteristic

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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 n0, 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)].

MSC

primary
18A25
secondary
20G05
20M25
16D90

Keywords

Representation theory
General linear groups
Finite fields
Functor categories

Cited by (0)

This research was partially supported by grants from the National Science Foundation.