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On the Hecke algebras and the colored HOMFLY polynomial

Authors: Xiao-Song Lin and Hao Zheng
Journal: Trans. Amer. Math. Soc. 362 (2010), 1-18
MSC (2000): Primary 57M27, 20C08
Published electronically: July 31, 2009
MathSciNet review: 2550143
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Abstract: The colored HOMFLY polynomial is the quantum invariant of oriented links in $ S^3$ associated with irreducible representations of the quantum group $ U_q(\mathrm{sl}_N)$. In this paper, using an approach to calculate quantum invariants of links via the cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mariño-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

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  • 1. A. K. Aiston and H. R. Morton, Idempotents of Hecke algebras of type A, J. Knot Theory Ramif. 7 (1998), 463-487. MR 1633027 (99h:57002)
  • 2. J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249-273. MR 992598 (90g:57004)
  • 3. R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20-52. MR 812444 (88b:20065)
  • 4. R. Dipper and G. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. 54 (1987), 57-82. MR 872250 (88m:20084)
  • 5. R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999), 1415-1443. MR 1796682 (2001k:81272)
  • 6. A. Gyoja, A q-analogue of Young symmetrizer, Osaka J. Math. 23 (1986), 841-852. MR 873212 (88e:20012)
  • 7. V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335-388. MR 908150 (89c:46092)
  • 8. C. Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer-Verlag, 1995. MR 1321145 (96e:17041)
  • 9. A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer-Verlag, 1997. MR 1492989 (99f:17017)
  • 10. J. M. F. Labastida and M. Mariño, Polynomial invariants for torus knots and topological strings, Comm. Math. Phys. 217 (2001), 423-449. MR 1821231 (2003b:57016)
  • 11. J. M. F. Labastida and M. Mariño, A new point of view in the theory of knot and link invariants, J. Knot Theory Ramif. 11 (2002), 173-197. MR 1895369 (2003h:57016)
  • 12. J. M. F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large N, J. High Energy Phys. 2000, no. 11, Paper 7, 42 pp. MR 1806596 (2003b:57015)
  • 13. R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), 1-94. MR 1427801 (98c:20015)
  • 14. H. Morton and S. Lukac, The Homfly polynomial of the decorated Hopf link, J. Knot Theory Ramif. 12 (2003), 395-416. MR 1983094 (2004d:57012)
  • 15. G. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), 492-513. MR 1194316 (94c:17031)
  • 16. H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000), 419-438. MR 1765411 (2001i:81254)
  • 17. N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26. MR 1036112 (91c:57016)
  • 18. B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth Inc., Belmont, California, 1991. MR 1093239 (93f:05102)
  • 19. V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527-553. MR 939474 (89e:57003)
  • 20. H. Wenzl, Quantum groups and subfactors of type B, C, and D, Comm. Math. Phys. 133 (1990), 383-432. MR 1090432 (92k:17032)
  • 21. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399. MR 990772 (90h:57009)

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Additional Information

Xiao-Song Lin
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Hao Zheng
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, People’s Republic of China

Received by editor(s): August 4, 2006
Published electronically: July 31, 2009
Additional Notes: The first author was supported in part by NSF grants DMS-0404511
The second author was supported in part by an NSFC grant
Article copyright: © Copyright 2009 American Mathematical Society

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