Polynomial representations of \(\mathrm{GL }(n)\) and Schur–Weyl duality

Original Paper

Abstract

Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur–Weyl duality is the key for an equivalence between both categories.

Keywords

Schur–Weyl duality Schur algebra Polynomial representation General linear group 

Mathematics Subject Classification

20G05 (20C30, 20G43) 

References

  1. Akin, K., Buchsbaum, D.A.: Characteristic-free representation theory of the general linear group. II. Homological considerations. Adv. Math. 72(2), 171–210 (1988)MathSciNetCrossRefGoogle Scholar
  2. Auslander, M.: Representation dimension of Artin algebras. Queen Mary College Mathematics Notes, London (1971)Google Scholar
  3. Benson, D., Doty, S.: Schur–Weyl duality over finite fields. Arch. Math. (Basel) 93(5), 425–435 (2009)MathSciNetCrossRefGoogle Scholar
  4. Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitres 4 à 7, Lecture Notes in Mathematics, 864, Masson, Paris (1981)Google Scholar
  5. Green, J.A.: Polynomial representations of \({\rm {GL}}_{n}\). Lecture Notes in Mathematics. Springer, Berlin (1980)MATHGoogle Scholar
  6. Macdonald, I.G.: Symmetric functions and Hall polynomials, second edition, Oxford Mathematical Monographs. Oxford Univ. Press, New York (1995)Google Scholar
  7. Schur, I.: Über die rationalen Darstellungen der allgemeinen linearen Gruppe, Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl., pp. 58–75, 1927. In: Schur, I., Gesammelte Abhandlungen III, 68–85, Springer, Berlin (1973)Google Scholar
  8. Stenström, B.: Rings of quotients. Springer, New York (1975)CrossRefGoogle Scholar

Copyright information

© The Managing Editors 2015

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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