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

Powell, Geoffrey M. L.

Trans. Am. Math. Soc. 350, No. 10, 4167-4193 (1998).

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.

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

Citations:

Zbl 0813.20049; Zbl 0805.55011; Zbl 0798.18009; Zbl 0871.55001

References:

[1]	E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85 – 99. · Zbl 0657.18005
[2]	Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders; Reprint of the 1981 original; A Wiley-Interscience Publication. · Zbl 0698.20001
[3]	S. Eilenberg and S. Mac Lane, On the groups \(H(\Pi, n)\), II, Ann. Math. 60, (1954), 49-139. · Zbl 0055.41704
[4]	Vincent Franjou, Jean Lannes, and Lionel Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994), no. 3, 513 – 538 (French, with English and French summaries). · Zbl 0798.18009
[5]	Eric M. Friedlander and Brian J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), no. 1, 235 – 253 (1986). · Zbl 0601.20042
[6]	Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323 – 448 (French). · Zbl 0201.35602
[7]	John C. Harris, Thomas J. Hunter, and R. James Shank, Steenrod algebra module maps from \?*(\?(\?/\?)\(^{n}\)) to \?*(\?(\?/\?)^{\?}), Proc. Amer. Math. Soc. 112 (1991), no. 1, 245 – 257. · Zbl 0722.55013
[8]	John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian \?-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427 – 449. · Zbl 0686.55007
[9]	Tsunekazu Kambe, Adams operation and \?-filtration in \?-theory, J. Math. Kyoto Univ. 32 (1992), no. 1, 93 – 100. · Zbl 0780.55012
[10]	Hans-Werner Henn, Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications, Math. Ann. 291 (1991), no. 2, 191 – 203. · Zbl 0751.55016
[11]	Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. J. Math. 115 (1993), no. 5, 1053 – 1106. · Zbl 0805.55011
[12]	Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. · Zbl 0654.20039
[13]	Piotr Krasoń and Nicholas J. Kuhn, On embedding polynomial functors in symmetric powers, J. Algebra 163 (1994), no. 1, 281 – 294. · Zbl 0832.20063
[14]	Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. I, Amer. J. Math. 116 (1994), no. 2, 327 – 360. · Zbl 0813.20049
[15]	Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II, \?-Theory 8 (1994), no. 4, 395 – 428. · Zbl 0830.20065
[16]	Nicholas J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. III, \?-Theory 9 (1995), no. 3, 273 – 303. · Zbl 0831.20057
[17]	Nicholas J. Kuhn, The Morava \?-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 (1987), no. 1, 193 – 205. · Zbl 0637.55002
[18]	Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un \?-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135 – 244 (French). With an appendix by Michel Zisman. · Zbl 0857.55011
[19]	Stuart Martin, Schur algebras and representation theory, Cambridge Tracts in Mathematics, vol. 112, Cambridge University Press, Cambridge, 1993. · Zbl 0802.20011
[20]	Stephen A. Mitchell and Stewart B. Priddy, Stable splittings derived from the Steinberg module, Topology 22 (1983), no. 3, 285 – 298. · Zbl 0526.55010
[21]	L. Piriou, Thèse de doctorat, Université de Paris VII, (1995).
[22]	G.M.L. Powell with an Appendix by L. Schwartz, The Artinian conjecture for \(I\otimes I\), To appear, J. Pure Appl. Alg.
[23]	G.M.L. Powell, Polynomial filtrations and Lannes’ \(T\)-functor, \(K\)-Theory 13 (1998), 279-304. CMP 98:10
[24]	G.M.L. Powell, The structure of \(\overline{I} \otimes \Lambda ^n\) in generic representation theory, J. Alg. 194 (1997), 455-465. CMP 97:17