Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-28T22:50:25.487Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Extract

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].

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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