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The structure of indecomposable injectives in generic representation theory. (English) Zbl 0903.18006

The goal of the paper under review is to study the structure of indecomposable injective objects in the (abelian) category \({\mathcal F}\) of functors from \({\mathcal E}_f\) to \({\mathcal E}\), where \({\mathcal E}\) is the category of \(\mathbb{F}\)-vector spaces, with \(\mathbb{F}\) a finite field, and \({\mathcal E}_f\) the subcategory of finite-dimensional spaces. The category \({\mathcal F}\) was called the category of generic representations by N. J. Kuhn [Am. J. Math. 116, No. 2, 327-360 (1994; Zbl 0813.20049)]. This category, which arised from topological motivations, was emphasized by Kuhn (see op. cit.), introduced by H.-W. Henn, J. Lannes and L. Schwartz [Am. J. Math. 115, No. 5, 1053-1106 (1993; Zbl 0805.55011)] for its relations with the category of unstable modules over the Steenrod algebra and studied from a homological point of view by V. Franjou, J. Lannes and L. Schwartz [Invent. Math. 115, No. 3, 513-538 (1994; Zbl 0798.18009)]. For his purpose, the author introduces the notions of “co-Weyl object” and of “\(J\)-good functor”.
A co-Weyl object is associated to each simple functor: if \(F\) is a simple functor then the corresponding co-Weyl object is the largest analytic functor which has simple socle \(F\) and is equal to \(F\) on \(\mathbb{F}^n\), where \(n\) is the smallest integer such that \(F\) is non zero on \(\mathbb{F}^n\) (the co-Weyl objects are related to the notion of Weyl ‘correspondence’ which is discussed in the section 5.8 of the book of L. Schwartz [“Unstable modules over Steenrod algebra and Sullivan’s fixed point set conjecture”, Chicago Lect. Math., Univ. of Chicago Press (1994; Zbl 0871.55001)]). Then a functor \(F\) in \({\mathcal F}\) is said to be \(J\)-good if it has a finite filtration of which the quotients are co-Weyl objects. Using this setting, the author proves that the indecomposable injective functors in \({\mathcal F}\) are \(J\)-good (meaning that the co-Weyl objects are the building blocks of the category \({\mathcal F})\), and as a corollary he shows that: The Cartan matrix \(\mathbb{F} [M_n]\) determines the Cartan matrix for \(\mathbb{F} [GL_n]\). In addition, for \(\mathbb{F}= \mathbb{F}_2\), the author gives a homological characterization of \(J\)-good functors and proves a “simplicity result” (section 6) for co-Weyl functors induced from a simple \(GL_n\)-module. Finally, the author discusses the category of Boolean valued functors and, if \(\mathbb{F}= \mathbb{F}_2\), the module structure for the co-Weyl objects. Furthermore, always in the case \(\mathbb{F}= \mathbb{F}_2\), he relates his results to the Artinian conjecture of Schwartz, Kuhn and Lannes.

MSC:

18G05 Projectives and injectives (category-theoretic aspects)
20G40 Linear algebraic groups over finite fields
55S10 Steenrod algebra
55U99 Applied homological algebra and category theory in algebraic topology
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