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An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

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Abstract

We develop several applications of the fact that the Yokonuma–Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori–Hecke algebras of type A . This includes a description of the semisimple and modular representation theory of the Yokonuma–Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism. In particular, for classical knots, a consequence of the construction is that the obtained set of invariants is topologically equivalent to the HOMFLYPT polynomial. We thus recover results of Chlouveraki et al. (2015, arXiv:1505.06666) about the Juyumaya–Lambropoulou invariants.

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Notes

  1. We thank G. Lusztig for pointing us this fact.

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Acknowledgments

The authors thank Cédric Bonnafé, Sofia Lambropoulou and Ivan Marin for useful discussions and comments. The authors are grateful to Georges Lusztig for having indicated to them the reference [18] which was not known at the time of the first version of this paper. The first author is supported by Agence National de la Recherche Projet ACORT ANR-12-JS01-0003.

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  1. Laboratoire de Mathématiques, UFR Sciences exactes et naturelles, Université de Reims Champagne-Ardenne, EA 4535 Moulin de la Housse, BP 1039, 51100, Reims, France

    N. Jacon & L. Poulain d’Andecy

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  1. N. Jacon
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  2. L. Poulain d’Andecy
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Correspondence to L. Poulain d’Andecy.

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Jacon, N., Poulain d’Andecy, L. An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants. Math. Z. 283, 301–338 (2016). https://doi.org/10.1007/s00209-015-1598-1

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