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ISSN 1079-6762

 
 

 

Generators and relations for Schur algebras


Authors: Stephen Doty and Anthony Giaquinto
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 54-62
MSC (2000): Primary 16P10, 16S15; Secondary 17B35, 17B37
DOI: https://doi.org/10.1090/S1079-6762-01-00094-4
Published electronically: June 26, 2001
MathSciNet review: 1852900
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Abstract | References | Similar Articles | Additional Information

Abstract:

We obtain a presentation of Schur algebras (and $q$-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra $\mathfrak{gl}_n$ of $n\times n$ matrices. We also find several new bases of Schur algebras.


References [Enhancements On Off] (What's this?)

  • 1. A.A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of ${\sf GL}_n$, Duke Math. J. 61 (1990), 655-677. MR 91m:17012
  • 2. S. Doty and A. Giaquinto, Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl}_2$, Algebras and Representation Theory, to appear.
  • 3. S. Doty and A. Giaquinto, Presenting quantum Schur algebras as quotients of the quantized enveloping algebra of $\mathfrak{gl}_2$, preprint, Loyola University Chicago, December 2000.
  • 4. S. Doty and A. Giaquinto, Presenting Schur algebras, preprint, Loyola University Chicago, April 2001.
  • 5. Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), 363-372. MR 96m:17024
  • 6. J. A. Green, Polynomial Representations of ${\sf GL}_n$ (Lecture Notes in Math. 830), Springer-Verlag, New York 1980. MR 83j:20003
  • 7. R. Green, $q$-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660-687. MR 97k:17016
  • 8. M. Jimbo, A $q$-analogue of $U(\mathfrak{gl}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Letters Math. Physics 11 (1986), 247-252. MR 87k:17011
  • 9. G. Lusztig, Introduction to Quantum Groups, Birkhäuser Boston 1993. MR 94m:17016
  • 10. H. Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), 349-383. MR 90b:46118

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Additional Information

Stephen Doty
Affiliation: Department of Mathematics, Loyola University, Chicago, IL 60626
Email: doty@math.luc.edu

Anthony Giaquinto
Affiliation: Department of Mathematics, Loyola University, Chicago, IL 60626
Email: tonyg@math.luc.edu

DOI: https://doi.org/10.1090/S1079-6762-01-00094-4
Keywords: Schur algebras, finite-dimensional algebras, enveloping algebras, quantized enveloping algebras
Received by editor(s): April 8, 2001
Published electronically: June 26, 2001
Communicated by: Alexandre Kirillov
Article copyright: © Copyright 2001 American Mathematical Society

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