Advertisement

Letters in Mathematical Physics

, Volume 30, Issue 3, pp 179–185 | Cite as

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

  • Paul Martin
  • Hubert Saleur
Article

Abstract

We determine the structure of the partition algebraPn(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.

Mathematics Subject Classifications (1991)

81R05 82B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 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. 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. 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. 5.
    Baxter, R. J.,Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.Google Scholar
  6. 6.
    Seleur, H.,Comm. Math. Phys. 132, 657 (1990); Kauffman, L. and Saleur, H.,Comm. Math. Phys. 152, 565 (1993).Google Scholar
  7. 7.
    Martin, P. P.,Publ. RIMS Kyoto Univ. 26, 485 (1991).Google Scholar
  8. 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. 9.
    MacDonald, I.G.,Symmetric Functions and Hall Polynomials, OUP, Oxford, 1979.Google Scholar
  10. 10.
    Liu, C. L.,Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968.Google Scholar
  11. 11.
    see, for example, Cohn, P.,Algebra vol. 2, Wiley, London, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Paul Martin
    • 1
  • Hubert Saleur
    • 2
  1. 1.Mathematics DepartmentCity UniversityLondonU.K.
  2. 2.Physics Department and Mathematics DepartmentUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations