Presenting Schur Algebras as Quotients of the Universal Enveloping Algebra of gl2
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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.
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- [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
- [Do]Donkin, S.: On Schur algebras and related algebras III: integral representations, Math. Proc. Cambridge Philos. Soc. 116 (1994), 37–55.Google Scholar
- [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
- [Gr]Green, J. A.: Polynomial Representations of GLn, Lecture Notes in Math. 830, Springer-Verlag, New York, 1980.Google Scholar
- [RG]Green, R.: q-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660–687.Google Scholar
- [Ja]Jantzen, J. C.: Representations of Algebraic Groups, Academic Press, Orlando, 1987.Google Scholar
- [Ko]Kostant, B.: Groups over ℤ, Proc. Sympos. Pure Math. 9 (1966), 90–98.Google Scholar