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

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.

💡 Research Summary

This paper continues the investigation of the functor category Fquad, which is built from finite‑dimensional vector spaces over the field F₂ equipped with a non‑degenerate quadratic form. The objects of Fquad are pairs (V,q) where V is an F₂‑vector space and q:V→F₂ is a non‑degenerate quadratic form; morphisms are linear maps preserving q. In earlier work the author introduced the standard projective objects P_V, defined by the Yoneda property Hom_{Fquad}(P_V, F) ≅ F(V) for any functor F.

The first major contribution of the present article is a new “degree filtration’’ on each standard projective P_V. For a given V, the filtration \