Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 1 (2017), 89-108.

Factorization of Temperley–Lieb diagrams

Dana C. Ernst, Michael G. Hastings, and Sarah K. Salmon

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The Temperley–Lieb algebra is a finite-dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type A. It is often realized in terms of a certain diagram algebra, where every diagram can be written as a product of “simple diagrams”. These factorizations correspond precisely to factorizations of the so-called fully commutative elements of the Coxeter group that index a particular basis. Given a reduced factorization of a fully commutative element, it is straightforward to construct the corresponding diagram. On the other hand, it is generally difficult to reconstruct the factorization given an arbitrary diagram. We present an efficient algorithm for obtaining a reduced factorization for a given diagram.

Article information

Source
Involve, Volume 10, Number 1 (2017), 89-108.

Dates
Received: 5 September 2015
Revised: 10 January 2016
Accepted: 14 January 2016
First available in Project Euclid: 22 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1511371085

Digital Object Identifier
doi:10.2140/involve.2017.10.89

Mathematical Reviews number (MathSciNet)
MR3561732

Zentralblatt MATH identifier
06642501

Subjects
Primary: 20C08: Hecke algebras and their representations 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 57M15: Relations with graph theory [See also 05Cxx]

Keywords
diagram algebra Temperley–Lieb algebra Coxeter group heap

Citation

Ernst, Dana C.; Hastings, Michael G.; Salmon, Sarah K. Factorization of Temperley–Lieb diagrams. Involve 10 (2017), no. 1, 89--108. doi:10.2140/involve.2017.10.89. https://projecteuclid.org/euclid.involve/1511371085


Export citation

References

  • S. C. Billey and B. C. Jones, “Embedded factor patterns for Deodhar elements in Kazhdan–Lusztig theory”, Ann. Comb. 11:3–4 (2007), 285–333.
    Mathematical Reviews (MathSciNet): MR2376108
    Zentralblatt MATH: 1189.20008
    Digital Object Identifier: doi:10.1007/s00026-007-0321-2
  • P. Cartier and D. Foata, Problèmes combinatoires de commutation et réarrangements, Lecture Notes in Mathematics 85, Springer, Berlin, 1969.
    Mathematical Reviews (MathSciNet): MR0239978
    Zentralblatt MATH: 0186.30101
  • D. C. Ernst, “Non-cancellable elements in type affine $C$ Coxeter groups”, Int. Electron. J. Algebra 8 (2010), 191–218.
    Mathematical Reviews (MathSciNet): MR2660550
    Zentralblatt MATH: 1253.20039
  • M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs $($N. S.$)$ 21, Oxford University Press, New York, NY, 2000.
    Mathematical Reviews (MathSciNet): MR1778802
    Zentralblatt MATH: 0996.20004
  • J. J. Graham, Modular representations of Hecke algebras and related algebras, thesis, University of Sydney, 1995.
  • R. M. Green, “Generalized Temperley–Lieb algebras and decorated tangles”, J. Knot Theory Ramifications 7:2 (1998), 155–171.
    Mathematical Reviews (MathSciNet): MR1618912
    Zentralblatt MATH: 0926.20005
    Digital Object Identifier: doi:10.1142/S0218216598000103
  • R. M. Green, “Star reducible Coxeter groups”, Glasg. Math. J. 48:3 (2006), 583–609.
    Mathematical Reviews (MathSciNet): MR2271387
    Zentralblatt MATH: 1149.20034
    Digital Object Identifier: doi:10.1017/S0017089506003211
  • J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990.
    Mathematical Reviews (MathSciNet): MR1066460
    Zentralblatt MATH: 0725.20028
  • V. F. R. Jones, “Planar algebras, I”, preprint, 1999.
    arXiv: math/9909027v1
  • L. H. Kauffman, “State models and the Jones polynomial”, Topology 26:3 (1987), 395–407.
    Mathematical Reviews (MathSciNet): MR899057
    Zentralblatt MATH: 0622.57004
    Digital Object Identifier: doi:10.1016/0040-9383(87)90009-7
  • L. H. Kauffman, “An invariant of regular isotopy”, Trans. Amer. Math. Soc. 318:2 (1990), 417–471.
    Mathematical Reviews (MathSciNet): MR958895
    Zentralblatt MATH: 0763.57004
    Digital Object Identifier: doi:10.1090/S0002-9947-1990-0958895-7
  • R. Penrose, “Angular momentum: an approach to combinatorial space-time”, pp. 151–180 in Quantum theory and beyond, edited by T. Bastin, Cambridge University Press, 1971.
  • J. R. Stembridge, “On the fully commutative elements of Coxeter groups”, J. Algebraic Combin. 5:4 (1996), 353–385.
    Mathematical Reviews (MathSciNet): MR1406459
    Zentralblatt MATH: 0864.20025
  • H. N. V. Temperley and E. H. Lieb, “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. Roy. Soc. London Ser. A 322:1549 (1971), 251–280.
    Mathematical Reviews (MathSciNet): MR0498284
    Zentralblatt MATH: 0211.56703
    Digital Object Identifier: doi:10.1098/rspa.1971.0067
  • G. X. Viennot, “Heaps of pieces, I: Basic definitions and combinatorial lemmas”, pp. 321–350 in Combinatoire énumérative (Montreal, QC, 1985), edited by G. Labelle and P. Leroux, Lecture Notes in Mathematics 1234, Springer, Berlin, 1986.
    Mathematical Reviews (MathSciNet): MR927773
    Zentralblatt MATH: 0618.05008

MSP

Involve: A Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more