The structure of indecomposable injectives in generic representation theory

Powell, Geoffrey

doi:10.1090/S0002-9947-98-02125-4

The structure of indecomposable injectives in generic representation theory
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by Geoffrey M. L. Powell

Trans. Amer. Math. Soc. 350 (1998), 4167-4193

DOI: https://doi.org/10.1090/S0002-9947-98-02125-4

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Abstract:

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.

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