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Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A

Author(s): Anthony Henderson
Journal: Represent. Theory 11 (2007), 95-121.
MSC (2000): Primary 17B37; Secondary 05E15, 20C08
Posted: June 26, 2007
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Abstract: The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and we derive some consequences for representation theory.

References:

1.
T. ARAKAWA, Drinfeld functor and finite-dimensional representations of Yangian, Commun. Math. Phys., 205 (1999), pp. 1-18. MR 1706920 (2001c:17011)

2.
S. ARIKI, On the decomposition number of the Hecke algebra of $ G(m,1,n)$, J. Math. Kyoto Univ., 36 (1996), pp. 789-808. MR 1443748 (98h:20012)

3.
S. BILLEY AND G. WARRINGTON, Maximal singular loci of Schubert varieties in $ SL(n)/B$, Trans. Amer. Math. Soc., 355 (2003), pp. 3915-3945. MR 1990570 (2004f:14071)

4.
A. BJÖRNER AND F. BRENTI, Affine permutations of type $ A$, Electron. J. Combin., 3 (1996).
Research Paper 18. MR 1392503 (97f:05018)

5.
V. CHARI AND A. PRESSLEY, A Guide to Quantum Groups, Cambridge Univ. Press, 1994. MR 1300632 (95j:17010)

6.
N. CHRISS AND V. GINZBURG, Representation Theory and Complex Geometry, Birkhäuser Boston, 1997. MR 1433132 (98i:22021)

7.
V. G. DRINFELD, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl., 20 (1986), pp. 67-70. MR 831053 (87m:22044)

8.
A. HENDERSON, Two-row nilpotent orbits of cyclic quivers, Math. Z., 243 (2003), pp. 127-143. MR 1953052 (2003k:16025)

9.
J. E. HUMPHREYS, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, 1990. MR 1066460 (92h:20002)

10.
B. LECLERC, J.-Y. THIBON, AND E. VASSEROT, Zelevinsky's involution at roots of unity, J. Reine Angew. Math., 513 (1999), pp. 33-51. MR 1713318 (2001f:20011)

11.
G. LUSZTIG, Affine Hecke algebras and their graded version, J. Amer. Math. Soc., 2 (1989), pp. 599-635. MR 991016 (90e:16049)

12.
-, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3 (1990), pp. 447-498. MR 1035415 (90m:17023)

13.
-, Cuspidal local systems and graded Hecke algebras II, in Representations of Groups (Banff, AB, 1994), no. 16 in CMS Conf. Proc., Amer. Math. Soc., 1995, pp. 217-275. MR 1357201 (96m:22038)

14.
-, Aperiodicity in quantum affine $ \mathfrak{gl}_n$, Asian J. Math., 3 (1999), pp. 147-177. MR 1701926 (2000i:17027)

15.
-, Bases in equivariant $ K$-theory II, Represent. Theory, 3 (1999), pp. 281-353. MR 1714628 (2000h:20085)

16.
M. NAZAROV AND V. TARASOV, Representations of Yangians with Gelfand-Zetlin bases, J. Reine Angew. Math., 496 (1998), pp. 181-212. MR 1605817 (99c:17030)

17.
R. ORELLANA AND A. RAM, Affine braids, Markov traces and the category $ \mathcal{O}$, to appear in the Proceedings of the International Congress 2004, held at Tata Institute of Fundamental Research.

18.
A. RAM, Skew shape representations are irreducible, in Combinatorial and Geometric Representation Theory (Seoul, 2001), no. 325 in Contemp. Math., Amer. Math. Soc., 2003, pp. 161-189. MR 1988991 (2004f:20014)

19.
J. D. ROGAWSKI, On modules over the Hecke algebra of a $ p$-adic group, Invent. Math., 79 (1985), pp. 443-465. MR 782228 (86j:22028)

20.
W. SOERGEL, Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Represent. Theory, 1 (1997), pp. 83-114. MR 1444322 (98d:17026)

21.
T. SUZUKI, Rogawski's conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type $ A$, Represent. Theory, 2 (1998), pp. 393-409. MR 1651408 (2000b:22016)

22.
M. VARAGNOLO AND E. VASSEROT, On the decomposition matrices of the quantized Schur algebra, Duke Math. J., 100 (1999), pp. 267-297. MR 1722955 (2001c:17029)

23.
E. VASSEROT, Affine quantum groups and equivariant $ K$-Theory, Transform. Groups, 3 (1998), pp. 269-299. MR 1640675 (99j:19007)

24.
A. ZELEVINSKY, Induced representations of reductive $ p$-adic groups II, Ann. Sci. Ecole Norm. Sup., 13 (1980), pp. 165-210. MR 584084 (83g:22012)

25.
-, $ p$-adic analogue of the Kazhdan-Lusztig hypothesis, Funct. Anal. Appl., 15 (1981), pp. 83-92.

26.
-, Two remarks on graded nilpotent classes, Russian Math. Surveys, 40 (1985), pp. 249-250. MR 783619 (86e:14027)

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Additional Information:

Anthony Henderson
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email: anthonyh@maths.usyd.edu.au

DOI: 10.1090/S1088-4165-07-00317-2
PII: S 1088-4165(07)00317-2
Received by editor(s): January 10, 2005
Posted: June 26, 2007
Additional Notes: This work was supported by Australian Research Council grant DP0344185
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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