Skip to main content
SpringerLink
Log in

Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We determine the structure of the partition algebraP n(Q) (a generalized Temperley-Lieb algebra) for specific values ofQ ∈ ℂ, focusing on the quotient which gives rise to the partition function ofn siteQ-state Potts models (in the continuousQ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is nonsemisimple iffQ is a nonnegative integer less than 2n-1. We determine the dimension of the key irreducible representation in every specialization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, G. E., Baxter, R. J., and Forrester, P. J.,J. Statist Phys. 35, 193 (1984); Huse, D. A.,Phys. Rev. B. 30, 3908 (1984); Belavin, A. A., Polyakov, and Zamolodchikov, A. B.,Nuclear Phys. B 241, 333 (1984); Date, E, Jimbo, M, Miwa, T., and Okado, M.,Nuclear Phys. B. 290 [FS20], 231 (1987); Deguchi, T., Wadati, M., and Akutsu, Y.,J. Phys. Soc. Japan,57, 2921 (1988); Pasquier, V. and Saleur, H.,Nuclear Phys. B 330, 523 (1990), and references therein.

    Google Scholar 

  2. Jones, V. F. R.,Invent. Math. 72, 1 (1983); Dipper, R. and James, G.,Proc. London Math. Soc. 52, 20 (1986) and references therein.

    Google Scholar 

  3. Cardy, J., in C. Domb and J. Lebowitz (eds),Phase Transitions and Critical Phenomena vol. 11, Academic Press, New York, 1987.

    Google Scholar 

  4. Martin, P. P., Non-planar statistical mechanics - The partition algebra construction, Yale preprint YCTP-P34-92, to appear inJ.K.T.R. Martin, P. P. and Saleur, H.,Comm. Math. Phys. 158, 155 (1993).

    Google Scholar 

  5. Baxter, R. J.,Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.

    Google Scholar 

  6. Seleur, H.,Comm. Math. Phys. 132, 657 (1990); Kauffman, L. and Saleur, H.,Comm. Math. Phys. 152, 565 (1993).

    Google Scholar 

  7. Martin, P. P.,Publ. RIMS Kyoto Univ. 26, 485 (1991).

    Google Scholar 

  8. Zamalodchikov, A. B.,Comm. Math. Phys. 79, 489 (1981); Baxter, R. J., in M. Rasettiet al. (eds),Integrable Systems in Statistical Mechanics, World Scientific, Singapore, 1985; Baxter, R. J. and Bazhanov, V. V.,J. Statist. Phys., to appear.

    Google Scholar 

  9. MacDonald, I.G.,Symmetric Functions and Hall Polynomials, OUP, Oxford, 1979.

    Google Scholar 

  10. Liu, C. L.,Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968.

    Google Scholar 

  11. see, for example, Cohn, P.,Algebra vol. 2, Wiley, London, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, City University, ECIV 0HB, London, U.K.

    Paul Martin

  2. Physics Department and Mathematics Department, University of Southern California, 90089, Los Angeles, CA, USA

    Hubert Saleur

Authors
  1. Paul Martin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hubert Saleur
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported by the Packard Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martin, P., Saleur, H. Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras. Lett Math Phys 30, 179–185 (1994). https://doi.org/10.1007/BF00805850

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00805850

Mathematics Subject Classifications (1991)

Search

Navigation