Abstract
In this paper, we describe the irreducible representations and give a dimension formula for the framisation of the Temperley–Lieb algebra. We then prove that the framisation of the Temperley–Lieb algebra is isomorphic to a direct sum of matrix algebras over tensor products of classical Temperley–Lieb algebras. This allows us to construct a basis for it. We also study in a similar way the complex reflection Temperley–Lieb algebra.
Similar content being viewed by others
References
Chlouveraki, M., Juyumaya, J., Karvounis, K., Lambropoulou, S.: Identifying the invariants for classical knots and links from the Yokonuma–Hecke algebras. arXiv:1505.06666
Chlouveraki, M., Lambropoulou, S.: The Yokonuma–Hecke algebras and the HOMFLYPT polynomial. J. Knot Theory Ramif. 22(14), 1350080 (2013)
Chlouveraki, M., Pouchin, G.: Determination of the representations and a basis for the Yokonuma–Temperley–Lieb algebra. Algebras Represent. Theory 18(2), 421–447 (2015)
Chlouveraki, M., Poulain d’Andecy, L.: Representation theory of the Yokonuma–Hecke algebra. Adv. Math. 259, 134–172 (2014)
Espinoza, J., Ryom-Hansen, S.: Cell Structures for the Yokonuma–Hecke Algebra and the Algebra of Braids and Ties. arXiv: 1506.00715
Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras. London Mathematical Society Monographs, New Series 21. Oxford University Press, New York (2000)
Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: The Yokonuma–Temperley–Lieb algebra. Banach Cent. Publ. 103, 24 (2014)
Goundaroulis, D., Juyumaya, J., Kontogeorgis, A., Lambropoulou, S.: Framization of the Temperley–Lieb algebra. Math. Res. Lett. arXiv:1304.7440
Hoefsmit, P.: Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type. PhD thesis, University of British Columbia (1974)
Jacon, N.: Poulain d’Andecy, L.: An isomorphism theorem for Yokonuma-Hecke algebras and applications to link invariants. Math. Z. 283(1), 301–338 (2016)
Jones, V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)
Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126(2), 335–388 (1987)
Juyumaya, J.: Sur les nouveaux générateurs de l’algèbre de Hecke H(G, U,1). J. Algebra 204, 49–68 (1998)
Juyumaya, J.: Markov trace on the Yokonuma–Hecke algebra. J. Knot Theory Ramif. 13, 25–39 (2004)
Juyumaya, J., Kannan, S.: Braid relations in the Yokonuma–Hecke algebra. J. Algebra 239, 272–297 (2001)
Juyumaya, J., Lambropoulou, S.: \(p\)-adic framed braids. Topol. Appl. 154, 1804–1826 (2007)
Juyumaya, J., Lambropoulou, S.: An invariant for singular knots. J. Knot Theory Ramif. 18(6), 825–840 (2009)
Juyumaya, J., Lambropoulou, S.: An adelic extension of the Jones polynomial. In: Banagl, M., Vogel, D. (eds.) The Mathematics of Knots, Contributions in the Mathematical and Computational Sciences, vol. 1. Springer, Berlin (2011)
Juyumaya, J., Lambropoulou, S.: \(p\)-adic framed braids II. Adv. Math. 234, 149–191 (2013)
Juyumaya, J., Lambropoulou, S.: On the framization of knot algebras. In: Kauffman, L.H., Manturov, V. (eds.) New Ideas in Low-dimensional Topology. Series of Knots and Everything, World Scientific Press (2014)
Lusztig, G.: Character sheaves on disconnected groups VII. Represent. Theory 9, 209–266 (2005)
Temperley, N., Lieb, E.: Relations between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation’ problem. Proc. R. Soc. Ser. A 322, 251–280 (1971)
Thiem, N.: Unipotent Hecke Algebras: The Structure, Representation Theory, and Combinatorics. Ph.D Thesis, University of Wisconsin (2004)
Thiem, N.: Unipotent Hecke algebras of \({{\rm GL}}_n({\mathbb{F}}_q)\). J. Algebra 284, 559–577 (2005)
Thiem, N.: A skein-like multiplication algorithm for unipotent Hecke algebras. Trans. Am. Math. Soc. 359(4), 1685–1724 (2007)
Yokonuma, T.: Sur la structure des anneaux de Hecke d’un groupe de Chevalley fini. C. R. Acad. Sci. Paris 264, 344–347 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
We are grateful to Dimoklis Goundaroulis, Jesús Juyumaya, Aristides Kontogeorgis and Sofia Lambropoulou for introducing us to this whole range of problems, and for many fruitful conversations. We would also like to thank Tamás Hausel for his interesting questions that led us to provide some extra results on the Yokonuma–Hecke algebra. Finally, we would like to thank Loïc Poulain d’Andecy and Nicolas Jacon for our useful conversations on the second part of this paper. This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS. The second author gratefully acknowledges financial support of EPSRC through the Grant Ep/I02610x/1.
Rights and permissions
About this article
Cite this article
Chlouveraki, M., Pouchin, G. Representation theory and an isomorphism theorem for the framisation of the Temperley–Lieb algebra. Math. Z. 285, 1357–1380 (2017). https://doi.org/10.1007/s00209-016-1751-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-016-1751-5