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On tilting modules for algebraic groups. (English) Zbl 0798.20035
Let $$G$$ be a reductive algebraic group over an algebraically closed field $$K$$ of characteristic $$p > 0$$. A rational $$G$$-module $$M$$ is called a (partial) tilting module if both $$M$$ and its dual $$M^*$$ may be filtered by modules $$\nabla(\lambda) = \text{Ind}^ G_ B K_ \lambda$$ ($$\lambda$$ a dominant weight), induced from one dimensional modules for a Borel subgroup $$B$$ of $$G$$. This paper is an investigation of tilting modules for $$G$$. There is an indecomposable tilting module $$M(\lambda)$$ for each dominant weight $$\lambda$$. Various results on tilting modules are established. In particular it is shown that the indecomposable tilting modules for $$G$$ behave well on truncation to a Levi subgroup of $$G$$. It is conjectured that one gets all the (restricted) principal indecomposable modules for the Lie algebra $$\mathfrak g$$ of $$G$$ from certain specified tilting modules by restricting the action from $$G$$ to $$\mathfrak g$$; and it is observed that this is true if the characteristic $$p$$ is at least $$2h - 2$$ (by results of Jantzen), where $$h$$ is the Coxeter number of $$G$$.
It is shown that the Ringel conjugate $$S(n,r)'$$ (defined by means of tilting modules) of the Schur algebra $$S(n,r)$$ is a “generalized Schur algebra” and in fact $$S(n,r)$$ is self conjugate when $$r \leq n$$. Two applications of this are given: a reciprocity formula stating that the filtration multiplicity of the induced module $$\nabla(\mu)$$ in the tilting module $$M(\lambda)$$ (in case $$G = GL(n)$$) is equal to a certain decomposition number; and a new proof of a result of Akin and Buchsbaum on Ext between induced modules (again with $$G = GL(n)$$). In a related paper by the author [Invent. Math. 110, No. 2, 389-401 (1992)] the theory of tilting modules as developed here is used to determine generators of the algebra of polynomial invariants for the action of the general linear group $$GL(n)$$, by simultaneous conjugation, on $$m$$-tuples of $$n \times n$$ matrices.
Reviewer: S.Donkin (London)

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16W20 Automorphisms and endomorphisms
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##### References:
 [1] Akin, K., Buchsbaum, D.A.: Characteristic-free representation theory of the general linear group II. Homological Considerations. Adv. Math.72, 171–210 (1988) · Zbl 0681.20028 · doi:10.1016/0001-8708(88)90027-8 [2] Alperin, J.L., Mason, G.: On simple modules for SL (2,q). Bull. Land. Math. Soc. (to appear) · Zbl 0820.20016 [3] Alperin, J.L., Mason, G.: Partial Steinberg modules for finite groups of Lie type. Bull. Land. Math. Soc. (to appear) · Zbl 0820.20017 [4] Ballard, J.W.: Injective modules for restricted enveloping algebras. Math. Z.163, 57–63 (1978) · Zbl 0378.17007 · doi:10.1007/BF01214444 [5] Cline, E., Parshall, B., Scott, L.L.: Algebraic stratification in representation categories. J. Algebra117, 504–521 (1988) · Zbl 0659.18011 · doi:10.1016/0021-8693(88)90123-8 [6] Cline, E., Parshall, B., Scott, L.L.: Finite dimensional algebras and highest weight categories. J. Reine Angew. Math.391, 85–99 (1988) · Zbl 0657.18005 [7] Cline, E., Parshall, B., Scott, L.L., van der Kallen, W.: Rational and generic cohomology. Invent. Math.39, 143–163 (1977) · Zbl 0346.20031 · doi:10.1007/BF01390106 [8] Collingwood, D.H., Irving, R.: A decomposition theorem for certain self-dual modules in the categoryO. Duke Math. J.58, 89–102 (1989) · Zbl 0673.17003 · doi:10.1215/S0012-7094-89-05806-7 [9] Donkin, S.: On a question of Verma. J. Lond. Math. Soc., Il. Ser.21, 445–455 (1980) · Zbl 0441.14014 · doi:10.1112/jlms/s2-21.3.445 [10] Donkin, S.: A filtration for rational modules. Math. Z.177, 1–8 (1981) · Zbl 0455.20029 · doi:10.1007/BF01214334 [11] Donkin, S.: A note on decomposition numbers for reductive algebraic groups. J. Algebra80, 226–234 (1983) · Zbl 0505.20028 · doi:10.1016/0021-8693(83)90029-7 [12] Donkin, S.: Rational Representations of Algebraic Groups: Tensor Products and Filtrations. (Lect. Notes Math., vol. 1140) Berlin Heidelberg New York: Springer 1985 · Zbl 0586.20017 [13] Donkin, S.: Finite resolutions of modules for reductive algebraic groups. J Algebra101, 473–488 (1986) · Zbl 0607.20023 · doi:10.1016/0021-8693(86)90206-1 [14] Donkin, S.: On Schur algebras and related algebras I. J. Algebra104, 310–328 (1986) · Zbl 0606.20038 · doi:10.1016/0021-8693(86)90218-8 [15] Donkin, S., On Schur algebras and related algebras II. J. Algebra111, 354–364 (1987) · Zbl 0634.20019 · doi:10.1016/0021-8693(87)90222-5 [16] Donkin, S.: Skew modules for reductive groups. J. Algebra113, 465–479 (1988) · Zbl 0662.20032 · doi:10.1016/0021-8693(88)90172-X [17] Donkin, S.: Invariants of several matrices. Invent. Math. (to appear) · Zbl 0826.20036 [18] Doty, S., Walker, G.: Modular symmetric functions and irreducible representations of general linear groups. J. Pure Appl. Algebra (to appear) · Zbl 0804.20034 [19] Erdmann, K.: Schur algebras of finite type. Q. J. Math. (to appear) · Zbl 0832.16011 [20] Grabmeier, J.: Unzerlegbare Moduln mit trivialer Youngquelle und Dastellungstheorie der Schuralgebra. Doctoral Thesis, University of Bayreuth (1985) · Zbl 0683.20015 [21] Green, J.A.: Polynomial Representations of GL n . (Lect. Notes Math., vol. 830) Berlin Heidelberg New York: Springer 1980 [22] Green, J.A.: Functor categories and group representations. Port. Math.43, 3–16 (1985–86) [23] Green, J.A.: A theorem on modular endomorphism rings. Ill. J. Math.32, 510–519 (1988) · Zbl 0659.16024 [24] Humphreys, J.E.: Ordinary and modular representations of Chevalley groups. (Lect. Notes Math., vol. 528) Berlin Heidelberg New York: Springer 1976 · Zbl 0341.20037 [25] Humphreys, J.E., Verma, D.-N.: Projective modules for finite Chevalley groups. Bull. Am. Math. Soc.79, 467–468 (1973) · Zbl 0258.20007 · doi:10.1090/S0002-9904-1973-13220-3 [26] James, G.D.: Trivial source modules for symmetric groups, Arch. Math.41, 294–300 (1983) · Zbl 0506.20004 · doi:10.1007/BF01371400 [27] Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und kontravariante Formen. J. Reine Angew. Math.290, 441–469 (1977) · Zbl 0342.20022 [28] Jantzen, J.C.: Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. J. Reine Angew. Math.317, 157–199 (1980) · Zbl 0451.20040 · doi:10.1515/crll.1980.317.157 [29] Jantzen, J.C.: Representations of Algebraic Groups. (Pure Appl. Math., vol. 131) New York London: Academic Press 1987 · Zbl 0654.20039 [30] Klyachko, A.A.: Direct summands of permutation modules. Sel. Math. Sov.3 (no. 1), 45–55 (1983/84) · Zbl 0588.20002 [31] Macdonald, I.G.: Symmetric functions and Hall polynomials. London, Oxford: Oxford University Press 1979 · Zbl 0487.20007 [32] Mathieu, O.: Filtrations ofG-modules. Ann. Sci. Éc. Norm. Super., II. Ser.23, 625–644 (1990) · Zbl 0748.20026 [33] Pillen, C.: Ph.D. Thesis, University of Massachusetts, Amherst (1992) [34] Ringel, C.M.: The category of modules with food filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z.208, 209–225 (1991) · Zbl 0725.16011 · doi:10.1007/BF02571521 [35] Suprunenko, I.D.: The invariance of the set of weights of irreducible representations of algebraic groups and Lie algebras of typeA 1 with restricted weights under reduction modulop (in Russian). Vesci. Akad. Navuk. BSSR, Ser. Fiz.-Mat.2, 18–22 (1983) · Zbl 0516.17002 [36] Upadhyaya, B.S.: Filtrations with Weyl module quotients of the principal indecomposable modules of the groups SL (2,q). Commun. Algebra7, 1469–1488 (1979) · Zbl 0425.20005 · doi:10.1080/00927877908822413 [37] Wang, J.-P.: Sheaf cohomology onG/B and tensor products of Weyl modules. J. Algebra77, 162–185 (1982) · Zbl 0493.20023 · doi:10.1016/0021-8693(82)90284-8
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