Tensor product varieties and crystals: case

Author:
Anton Malkin

Journal:
Trans. Amer. Math. Soc. **354** (2002), 675-704

MSC (2000):
Primary 20G99, 14M15

DOI:
https://doi.org/10.1090/S0002-9947-01-02899-9

Published electronically:
October 3, 2001

MathSciNet review:
1862563

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A geometric theory of tensor product for -crystals is described. In particular, the role of Spaltenstein varieties in the tensor product is explained, and thus a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.

**[BM83]**W. Borho and R. MacPherson,*Partial resolutions of nilpotent varieties*, Astérisque,**101-102**(1983), pp. 23-74. MR**85j:14087****[Gin91]**V. Ginzburg,*Lagrangian construction of the enveloping algebra*, C. R. Acad. Sci. Paris Sér. I Math.**312**(1991), no. 12, 907-912. MR**92c:17017****[GL92]**I. Grojnowski and G. Lusztig,*On bases of irreducible representations of quantum*, Kazhdan-Lusztig theory and related topics, Contemp. Math.**139**(Chicago, IL, 1989), AMS, Providence, RI, 1992, pp. 167-174. MR**94a:20070****[Hal59]**P. Hall,*The algebra of partitions*, Proc. 4th Canadian Math. Congress (Banff), University of Toronto Press, 1959, pp. 147-159. MR**28:1074****[Jos95]**A. Joseph,*Quantum groups and their primitive ideals*, Springer-Verlag, Berlin, 1995. MR**96d:17015****[Kas90]**M. Kashiwara,*Crystalizing the -analogue of universal enveloping algebras*, Comm. Math. Phys.**133**(1990), no. 2, 249-260.**[Kas91]**M. Kashiwara,*On crystal bases of the -analogue of universal enveloping algebras*, Duke Math. J.**63**(1991), no. 2, 465-516. MR**93b:17045****[Kas94]**M. Kashiwara,*Crystal bases of modified quantized enveloping algebra*, Duke Math. J.**73**(1994), no. 2, 383-413. MR**95c:17024****[KS97]**M. Kashiwara and Y. Saito,*Geometric construction of crystal bases*, Duke Math. J.**89**(1997), no. 1, 9-36. MR**99e:17025****[Lus91]**G. Lusztig,*Quivers, perverse sheaves, and quantized enveloping algebras*, J. Amer. Math. Soc.**4**(1991), no. 2, 365-421. MR**91m:17018****[LW54]**S. Lang and A. Weil,*Number of points of varieties in finite fields*, Amer. J. Math.**76**(1954), 819-827. MR**16:398d****[Mac95]**I. Macdonald,*Symmetric functions and Hall polynomials*, second ed., The Clarendon Press Oxford University Press, New York, 1995. MR**96h:05207****[Mal01]**A. Malkin,*Tensor product varieties and crystals. ADE case.*, arXiv:math.AG/0103025**[Nak94]**H. Nakajima,*Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras*, Duke Math. J.**76**(1994), no. 2, 365-416.**[Nak98]**H. Nakajima,*Quiver varieties and Kac-Moody algebras*, Duke Math. J.**91**(1998), no. 3, 515-560. MR**99b:17033****[Nak01]**H. Nakajima,*Quiver varieties and tensor products*, arXiv:math.QA/0103008**[Spa82]**N. Spaltenstein,*Classes unipotentes et sous-groupes de Borel*, Springer-Verlag, Berlin, 1982. MR**84a:14024**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20G99,
14M15

Retrieve articles in all journals with MSC (2000): 20G99, 14M15

Additional Information

**Anton Malkin**

Affiliation:
Department of Mathematics, Yale University, P.O. Box 208283, New Haven, Connecticut 06520-8283

Address at time of publication:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

Email:
malkin@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02899-9

Received by editor(s):
March 7, 2001

Published electronically:
October 3, 2001

Article copyright:
© Copyright 2001
American Mathematical Society