# D\'ecompositions \`a la Steinberg sur une cat\'egorie additive

## TL;DR

This paper characterizes simple functors with finitely generated values in additive categories, extending Steinberg's tensor product theorems, with applications to general linear group representations and functor category properties.

## Contribution

It provides a Steinberg-like decomposition for polynomial functors in additive categories, connecting representation theory and functor categories.

## Key findings

- Description of simple functors with finitely generated values
- Extension of Steinberg's tensor product theorems
- Applications to general linear group representations

## Abstract

We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation theory. Our results rest on the notion of polynomial functor introduced by Eilenberg and Mac Lane. We give applications to representations of general linear groups or to finiteness properties of functor categories.