Algebras and Representation Theory

, Volume 7, Issue 1, pp 1–17 | Cite as

Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2

  • Stephen Doty
  • Anthony Giaquinto


We give a presentation of the Schur algebras S Q (2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S Q (2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis.

Schur algebra enveloping algebra generators and relations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CL]
    Carter, R. and Lusztig, G.: On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.Google Scholar
  2. [Do]
    Donkin, S.: On Schur algebras and related algebras III: integral representations, Math. Proc. Cambridge Philos. Soc. 116 (1994), 37–55.Google Scholar
  3. [DG]
    Doty, S. R. and Giaquinto, A.: Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl2, Preprint, Loyola Univ. Chicago, Sept. 2000.Google Scholar
  4. [Gr]
    Green, J. A.: Polynomial Representations of GLn, Lecture Notes in Math. 830, Springer-Verlag, New York, 1980.Google Scholar
  5. [RG]
    Green, R.: q-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660–687.Google Scholar
  6. [Ja]
    Jantzen, J. C.: Representations of Algebraic Groups, Academic Press, Orlando, 1987.Google Scholar
  7. [Ko]
    Kostant, B.: Groups over ℤ, Proc. Sympos. Pure Math. 9 (1966), 90–98.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Stephen Doty
    • 1
  • Anthony Giaquinto
    • 1
  1. 1.Mathematical and Computer SciencesLoyola University ChicagoChicagoU.S.A.

Personalised recommendations