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On Schur algebras and related algebras III: integral representations

Published online by Cambridge University Press:  24 October 2008

Stephen Donkin
Stephen Donkin
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS
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Let G be a reductive group over an algebraically closed field K. In [8] and [9] we defined and studied certain finite dimensional K-algebras SK(π), associated to G via a finite saturated set π of dominant weights. The algebras are defined over ℤ, i.e. SK(π) = K ⊗ℤSℤ(π) for an order Sℤ(π) of Sℚ(π), and if G is a general linear group or a Chevalley group then the order Sℤ(π) arises naturally from the corresponding group scheme G over ℤ (or Kostant ℤ-form Uℤ). These algebras may be regarded as (and were obtained as) direct generalizations of the Schur algebras S(n, r) studied by Green in [10].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 1 , July 1994 , pp. 37 - 55
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Ballard, J. W.. Injective modules for restricted enveloping algebras. Math. Zeit. 163 (1978), 56–63.CrossRefGoogle Scholar
[2]Cline, E., Parshall, B. and Scott, L. L.. Cohomology, hyperalgebras and representations. J. Algebra 63 (1980), 98–123.CrossRefGoogle Scholar
[3]Curtis, C. W. and Reiner, I.. Representation theory of finite groups and associative algebras (Wiley, 1966).Google Scholar
[4]Curtis, C. W. and Reiner, I.. Methods of Representation Theory I (Wiley, 1981).Google Scholar
[5]Curtis, C. W. and Reiner, I., Methods of Representation Theory II (Wiley, 1987).Google Scholar
[6]Donkin, S.. Hopf complements and injective comodules. Proc. Lond. Math. Soc. (3) 40 (1980), 298–319.CrossRefGoogle Scholar
[7]Donkin, S.. A filtration for rational modules. Math. Zeit. 111 (1981), 1–8.CrossRefGoogle Scholar
[8]Donkin, S.. On Schur algebras and related algebras I. J. Algebra 104 (1986), 310–328.CrossRefGoogle Scholar
[9]Donkin, S.. On Schur algebras and related algebras II. J. Algebra 111 (1987), 354–364.CrossRefGoogle Scholar
[10]Green, J. A.. Polynomial Representations of GLn. Lecture Notes in Mathematics, vol. 830 (Springer, 1980).Google Scholar
[11]Humphreys, J. E.. Lie Algebras and Representation Theory. Graduate Text in Mathematics, vol. 9 (Springer, 1972).Google Scholar
[12]Humphreys, J. E.. Ordinary and Modular Representations of Chevalley Groups. Lecture Notes in Mathematics, vol. 528 (Springer, 1976).CrossRefGoogle Scholar
[13]Jantzen, J. C.. Darstellungen halbeinfacher Gruppen und kontravarianten Formen. J. reine angew. Math. 290 (1977), 441–469.Google Scholar
[14]Jantzen, J. C.. Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne. J. reine angew. Math. 317 (1980), 157–199.Google Scholar
[15]Jantzen, J. C.. Representations of Algebraic Groups. Pure and Applied Mathematics, vol. 131 (Academic Press, 1987).Google Scholar
[16]Lusztig, G.. Divisibility of projective modules of finite Chevalley groups by the Steinberg module. Bull. London Math. Soc. 8 (1976), 130–134.CrossRefGoogle Scholar
[17]McConnell, J. C.. The K-Theory of Filtered Rings and Skew Laurent Extensions. In: Séminaire d'Algèbre P.Dubreil et M.-P.Malliavin, Proc. 1983–84. Lecture Notes in Mathematics, vol. 1146 (Springer, 1985).Google Scholar
[18]Nichols, W. D. and Zoeller, M. B.. A Hopf algebra freeness theorem. Amer. J. Math. 111 (1989), 381–385.CrossRefGoogle Scholar
[19]Steinberg, R.. Representations of algebraic groups. Nagoya Math. J. 22 (1963), 33–56.CrossRefGoogle Scholar
[20]Sullivan, J. B.. Simply connected groups, the hyperalgebra and Verma's conjecture. Amer. J. Math. 102 (1978), 1015–1019.CrossRefGoogle Scholar
[21]Taylor, M. J.. The locally free classgroup of the symmetric group. Ill. J. Math. 23 (1979), 687–702.Google Scholar