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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.

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.

Reviewer: Philippe Elbaz-Vincent (Montpellier)

### 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 |

### Keywords:

Artinian object; polynomial filtration; shift functor; indecomposable injective objects; category of generic representations; Steenrod algebra; simple functor; analytic functor; co-Weyl objects; \(J\)-good functors; Artinian conjecture
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\textit{G. M. L. Powell}, Trans. Am. Math. Soc. 350, No. 10, 4167--4193 (1998; Zbl 0903.18006)

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### References:

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