# Generic representations of orthogonal groups: projective functors in the category Fquad

## TL;DR

This paper advances the understanding of the category Fquad, associated with quadratic forms over F_2, by decomposing projective objects and characterizing polynomial functors, thus enriching the algebraic structure analysis.

## Contribution

It introduces a filtration and decomposition of standard projective objects in Fquad, providing a detailed description of polynomial functors within this category.

## Key findings

- Decomposition of the first two standard projective objects in Fquad.
- Complete description of polynomial functors in Fquad.
- Refined understanding of the algebraic structure of Fquad.

## Abstract

In this paper, we continue the study of the category of functors Fquad, associated to F_2-vector spaces equipped with a nondegenerate quadratic form, initiated in two previous papers of the author. We define a filtration of the standard projective objects in Fquad; this refines to give a decomposition into indecomposable factors of the two first standard projective objects in Fquad. As an application of these two decompositions, we give a complete description of the polynomial functors of the category Fquad.