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

HTML articles powered by AMS MathViewer

- by Anthony Henderson PDF
- Represent. Theory
**11**(2007), 95-121 Request permission

## 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

- Tomoyuki Arakawa,
*Drinfeld functor and finite-dimensional representations of Yangian*, Comm. Math. Phys.**205**(1999), no. 1, 1–18. MR**1706920**, DOI 10.1007/s002200050664 - Susumu Ariki,
*On the decomposition numbers of the Hecke algebra of $G(m,1,n)$*, J. Math. Kyoto Univ.**36**(1996), no. 4, 789–808. MR**1443748**, DOI 10.1215/kjm/1250518452 - Sara C. Billey and Gregory S. Warrington,
*Maximal singular loci of Schubert varieties in $\textrm {SL}(n)/B$*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 3915–3945. MR**1990570**, DOI 10.1090/S0002-9947-03-03019-8 - Anders Björner and Francesco Brenti,
*Affine permutations of type $A$*, Electron. J. Combin.**3**(1996), no. 2, Research Paper 18, approx. 35. The Foata Festschrift. MR**1392503** - Vyjayanthi Chari and Andrew Pressley,
*A guide to quantum groups*, Cambridge University Press, Cambridge, 1994. MR**1300632** - Neil Chriss and Victor Ginzburg,
*Representation theory and complex geometry*, Birkhäuser Boston, Inc., Boston, MA, 1997. MR**1433132** - V. G. Drinfel′d,
*Degenerate affine Hecke algebras and Yangians*, Funktsional. Anal. i Prilozhen.**20**(1986), no. 1, 69–70 (Russian). MR**831053** - Anthony Henderson,
*Two-row nilpotent orbits of cyclic quivers*, Math. Z.**243**(2003), no. 1, 127–143. MR**1953052**, DOI 10.1007/s00209-002-0455-1 - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460**, DOI 10.1017/CBO9780511623646 - Bernard Leclerc, Jean-Yves Thibon, and Eric Vasserot,
*Zelevinsky’s involution at roots of unity*, J. Reine Angew. Math.**513**(1999), 33–51. MR**1713318**, DOI 10.1515/crll.1999.062 - George Lusztig,
*Affine Hecke algebras and their graded version*, J. Amer. Math. Soc.**2**(1989), no. 3, 599–635. MR**991016**, DOI 10.1090/S0894-0347-1989-0991016-9 - G. Lusztig,
*Canonical bases arising from quantized enveloping algebras*, J. Amer. Math. Soc.**3**(1990), no. 2, 447–498. MR**1035415**, DOI 10.1090/S0894-0347-1990-1035415-6 - George Lusztig,
*Cuspidal local systems and graded Hecke algebras. II*, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 217–275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 145–202; MR0972345 (90e:22029)]. MR**1357201**, DOI 10.1090/S1088-4165-02-00172-3 - G. Lusztig,
*Aperiodicity in quantum affine $\mathfrak {g}\mathfrak {l}_n$*, Asian J. Math.**3**(1999), no. 1, 147–177. Sir Michael Atiyah: a great mathematician of the twentieth century. MR**1701926**, DOI 10.4310/AJM.1999.v3.n1.a7 - G. Lusztig,
*Bases in equivariant $K$-theory. II*, Represent. Theory**3**(1999), 281–353. MR**1714628**, DOI 10.1090/S1088-4165-99-00083-7 - Maxim Nazarov and Vitaly Tarasov,
*Representations of Yangians with Gelfand-Zetlin bases*, J. Reine Angew. Math.**496**(1998), 181–212. MR**1605817**, DOI 10.1515/crll.1998.029 - 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. - Arun Ram,
*Skew shape representations are irreducible*, Combinatorial and geometric representation theory (Seoul, 2001) Contemp. Math., vol. 325, Amer. Math. Soc., Providence, RI, 2003, pp. 161–189. MR**1988991**, DOI 10.1090/conm/325/05670 - J. D. Rogawski,
*On modules over the Hecke algebra of a $p$-adic group*, Invent. Math.**79**(1985), no. 3, 443–465. MR**782228**, DOI 10.1007/BF01388516 - Wolfgang Soergel,
*Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules*, Represent. Theory**1**(1997), 83–114. MR**1444322**, DOI 10.1090/S1088-4165-97-00021-6 - Takeshi Suzuki,
*Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type $A$*, Represent. Theory**2**(1998), 393–409. MR**1651408**, DOI 10.1090/S1088-4165-98-00043-0 - Michela Varagnolo and Eric Vasserot,
*On the decomposition matrices of the quantized Schur algebra*, Duke Math. J.**100**(1999), no. 2, 267–297. MR**1722955**, DOI 10.1215/S0012-7094-99-10010-X - E. Vasserot,
*Affine quantum groups and equivariant $K$-theory*, Transform. Groups**3**(1998), no. 3, 269–299. MR**1640675**, DOI 10.1007/BF01236876 - A. V. Zelevinsky,
*Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$*, Ann. Sci. École Norm. Sup. (4)**13**(1980), no. 2, 165–210. MR**584084**, DOI 10.24033/asens.1379 - —,
*$p$-adic analogue of the Kazhdan-Lusztig hypothesis*, Funct. Anal. Appl., 15 (1981), pp. 83–92. - A. V. Zelevinskiĭ,
*Two remarks on graded nilpotent classes*, Uspekhi Mat. Nauk**40**(1985), no. 1(241), 199–200 (Russian). MR**783619**

## Additional Information

**Anthony Henderson**- Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
- MR Author ID: 687061
- ORCID: 0000-0002-3965-7259
- Email: anthonyh@maths.usyd.edu.au
- Received by editor(s): January 10, 2005
- Published electronically: June 26, 2007
- Additional Notes: This work was supported by Australian Research Council grant DP0344185
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**11**(2007), 95-121 - MSC (2000): Primary 17B37; Secondary 05E15, 20C08
- DOI: https://doi.org/10.1090/S1088-4165-07-00317-2
- MathSciNet review: 2320806