The structure of indecomposable injectives in generic representation theory

This paper considers the structure of the injective objects $I_{V_{n}}$ in the category $\mathcal F$ of functors between ${\mathbb F}_2$-vector spaces. A co-Weyl object $J_\lambda$ is defined, for each simple functor $F_\lambda$ in $\mathcal F$. A functor is defined to be $J$-good if it admits a finite filtration of which the quotients are co-Weyl objects. Properties of $J$-good functors are considered and it is shown that the indecomposable injectives in $\mathcal F$ are $J$-good. A finiteness result for proper sub-functors of co-Weyl objects is proven, using the polynomial filtration of the shift functor $\tilde\Delta: \mathcal F \rightarrow \mathcal F$. This research is motivated by the Artinian conjecture due to Kuhn, Lannes and Schwartz.