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On the Hecke algebras and the colored HOMFLY polynomial. (English) Zbl 1193.57006

This paper offers a rigorous mathematical definition of the coloured HOMFLY polynomial (Definition 4.4) and a robust mathematical formulation of the LMOV (Labastida-Mariño-Ooguri-Vafa) Conjecture (Conjecture 6.1). This conjecture, a remarkable consequence of large N Chern-Simons/topological string duality, is verified in several lower degree cases for some torus links (Section 6).
The coloured HOMFLY polynomial is calculated for torus links using a standard idea: fusing strands using representation theory, then exploiting known eigenvalues of the twist operators (the “Cabling Projection Rule”, Lemma 3.3). The calculation boils down to determining certain generalizations of Littlewood-Richardson coefficients, related to characters of Hecke algebras and Schur polynomials.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
20C08 Hecke algebras and their representations
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] A. K. Aiston and H. R. Morton, Idempotents of Hecke algebras of type \?, J. Knot Theory Ramifications 7 (1998), no. 4, 463 – 487. · Zbl 0924.57005 · doi:10.1142/S0218216598000243
[2] Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249 – 273. · Zbl 0684.57004
[3] Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20 – 52. · Zbl 0587.20007 · doi:10.1112/plms/s3-52.1.20
[4] Richard Dipper and Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57 – 82. · Zbl 0615.20009 · doi:10.1112/plms/s3-54.1.57
[5] Rajesh Gopakumar and Cumrun Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999), no. 5, 1415 – 1443. · Zbl 0972.81135 · doi:10.4310/ATMP.1999.v3.n5.a5
[6] Akihiko Gyoja, A \?-analogue of Young symmetrizer, Osaka J. Math. 23 (1986), no. 4, 841 – 852. · Zbl 0644.20012
[7] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335 – 388. · Zbl 0631.57005 · doi:10.2307/1971403
[8] Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. · Zbl 0808.17003
[9] Anatoli Klimyk and Konrad Schmüdgen, Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. · Zbl 0891.17010
[10] J. M. F. Labastida and Marcos Mariño, Polynomial invariants for torus knots and topological strings, Comm. Math. Phys. 217 (2001), no. 2, 423 – 449. · Zbl 1018.81049 · doi:10.1007/s002200100374
[11] José M. F. Labastida and Marcos Mariño, A new point of view in the theory of knot and link invariants, J. Knot Theory Ramifications 11 (2002), no. 2, 173 – 197. · Zbl 1002.57026 · doi:10.1142/S0218216502001561
[12] José M. F. Labastida, Marcos Mariño, and Cumrun Vafa, Knots, links and branes at large \?, J. High Energy Phys. 11 (2000), Paper 7, 42. · Zbl 0990.81545 · doi:10.1088/1126-6708/2000/11/007
[13] Robert Leduc and Arun 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), no. 1, 1 – 94. · Zbl 0936.17016 · doi:10.1006/aima.1997.1602
[14] Hugh R. Morton and Sascha G. Lukac, The Homfly polynomial of the decorated Hopf link, J. Knot Theory Ramifications 12 (2003), no. 3, 395 – 416. · Zbl 1055.57014 · doi:10.1142/S0218216503002536
[15] G. E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), no. 2, 492 – 513. · Zbl 0794.20020 · doi:10.1016/0021-8693(92)90045-N
[16] Hirosi Ooguri and Cumrun Vafa, Knot invariants and topological strings, Nuclear Phys. B 577 (2000), no. 3, 419 – 438. · Zbl 1036.81515 · doi:10.1016/S0550-3213(00)00118-8
[17] N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1 – 26. · Zbl 0768.57003
[18] Bruce E. Sagan, The symmetric group, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991. Representations, combinatorial algorithms, and symmetric functions. · Zbl 0823.05061
[19] V. G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), no. 3, 527 – 553. · Zbl 0648.57003 · doi:10.1007/BF01393746
[20] Hans Wenzl, Quantum groups and subfactors of type \?, \?, and \?, Comm. Math. Phys. 133 (1990), no. 2, 383 – 432. · Zbl 0744.17021
[21] Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351 – 399. · Zbl 0667.57005
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