Remarks on \(\boldsymbol {A^{(1)}_n}\) Face Weights

  • Atsuo Kuniba
Part of the MATRIX Book Series book series (MXBS, volume 2)


Elementary proofs are presented for the factorization of the elliptic Boltzmann weights of the \(A^{(1)}_n\) face model, and for the sum-to-1 property in the trigonometric limit, at a special point of the spectral parameter. They generalize recent results obtained in the context of the corresponding trigonometric vertex model.


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The author thanks Masato Okado for discussion. This work is supported by Grants-in-Aid for Scientific Research No. 15K13429 from JSPS.


  1. 1.
    Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys. 35, 193–266 (1984)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982)zbMATHGoogle Scholar
  3. 3.
    Borodin, A.: Symmetric elliptic functions, IRF models, and dynamic exclusion processes (2017). arXiv:1701.05239Google Scholar
  4. 4.
    Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models II: proof of the star-triangle relation and combinatorial identities. Adv. Stud. Pure Math. 16, 17–122 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jimbo, M., Miwa, T., Okado, M.: Symmetric tensors of the \(A^{(1)}_{n-1}\) family. Algebr. Anal. 1, 253–266 (1988)Google Scholar
  6. 6.
    Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: The \(A^{(1)}_n\) face models. Commun. Math. Phys. 119, 543–565 (1989)Google Scholar
  7. 7.
    Kuan, J.: An algebraic construction of duality functions for the stochastic \(U_q(A_n^{(1)})\) vertex model and its degenerations (2017). arXiv:1701.04468Google Scholar
  8. 8.
    Kuniba, A., Mangazeev, V.V., Maruyama, S., Okado, M.: Stochastic R matrix for \(U_q(A^{(1)}_n)\). Nucl. Phys. B 913, 248–277 (2016)Google Scholar

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Authors and Affiliations

  • Atsuo Kuniba
    • 1
  1. 1.Institute of PhysicsUniversity of TokyoKomaba, TokyoJapan

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