Division functors and structure of \(I^{\otimes 2}\otimes \Lambda^n\) in the category \(\mathcal F\). (Foncteurs de division et structure de \(I^{\otimes 2}\otimes \Lambda^n\) dans la catégorie \(\mathcal F\).) (French. English summary) Zbl 1132.18002

Author’s summary: We prove that, in the category \(\mathcal F\) of functors between \({\mathbb F}_2\)-vector spaces, the tensor product between the second non-constant standard injective functors \(V\mapsto {\mathbb F}_2^{(V^*)^{\oplus 2}}\) and an exterior power functor is Artinian. The only case known to date was the artin character of this injective; our result is a step in the study of the third non-constant standard injective of \(\mathcal F\). We use the division functor by the identity functor and facts from modular representation theory of the symmetric groups to obtain this theorem by detecting suitable composition factors.


18A25 Functor categories, comma categories
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
20B30 Symmetric groups
20C20 Modular representations and characters
55S10 Steenrod algebra
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