Howe duality and the quantum general linear group

Author:
R. B. Zhang

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2681-2692

MSC (2000):
Primary 17B37, 20G42, 17B10

Published electronically:
December 30, 2002

MathSciNet review:
1974323

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Abstract | References | Similar Articles | Additional Information

Abstract: A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo's duality between and the Hecke algebra.

**[APW]**Henning Haahr Andersen, Patrick Polo, and Ke Xin Wen,*Representations of quantum algebras*, Invent. Math.**104**(1991), no. 1, 1–59. MR**1094046**, 10.1007/BF01245066**[CP]**Vyjayanthi Chari and Andrew Pressley,*A guide to quantum groups*, Cambridge University Press, Cambridge, 1994. MR**1300632****[GZ]**A. R. Gover and R. B. Zhang,*Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I*, Rev. Math. Phys.**11**(1999), no. 5, 533–552. MR**1696104**, 10.1142/S0129055X99000209**[Ho]**Roger Howe,*Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond*, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR**1321638**, 10.1007/BF02771542**[Ji]**Michio Jimbo,*Quantum 𝑅 matrix related to the generalized Toda system: an algebraic approach*, Field theory, quantum gravity and strings (Meudon/Paris, 1984/1985), Lecture Notes in Phys., vol. 246, Springer, Berlin, 1986, pp. 335–361. MR**848629**, 10.1007/3-540-16452-9_21**[Ko]**Bertram Kostant,*On Macdonald’s 𝜂-function formula, the Laplacian and generalized exponents*, Advances in Math.**20**(1976), no. 2, 179–212. MR**0485661****[KR]**A. N. Kirillov and N. Reshetikhin,*𝑞-Weyl group and a multiplicative formula for universal 𝑅-matrices*, Comm. Math. Phys.**134**(1990), no. 2, 421–431. MR**1081014****[Ma]**Andrew Mathas,*Iwahori-Hecke algebras and Schur algebras of the symmetric group*, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR**1711316****[Mon]**Susan Montgomery,*Hopf algebras and their actions on rings*, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR**1243637****[PW]**Brian Parshall and Jian Pan Wang,*Quantum linear groups*, Mem. Amer. Math. Soc.**89**(1991), no. 439, vi+157. MR**1048073**, 10.1090/memo/0439**[FRT]**N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev,*Quantization of Lie groups and Lie algebras*, Algebra i Analiz**1**(1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J.**1**(1990), no. 1, 193–225. MR**1015339****[Ta]**Mitsuhiro Takeuchi,*Some topics on 𝐺𝐿_{𝑞}(𝑛)*, J. Algebra**147**(1992), no. 2, 379–410. MR**1161300**, 10.1016/0021-8693(92)90212-5

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Additional Information

**R. B. Zhang**

Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales 2006, Australia

Email:
rzhang@maths.usyd.edu.au

DOI:
https://doi.org/10.1090/S0002-9939-02-06892-2

Received by editor(s):
June 24, 2001

Received by editor(s) in revised form:
April 7, 2002

Published electronically:
December 30, 2002

Communicated by:
Dan M. Barbasch

Article copyright:
© Copyright 2002
American Mathematical Society