The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials

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• 作者：Tiago Fonseca (1) (2)
  Ferenc Balogh (3) (4)
  
  1. Centre de Recherches Math茅matiques ; Universit茅 de Montr茅al ; Montr茅al ; QC ; Canada
  2. LAPTh and CNRS ; 9 chemin de Bellevue ; BP 110 ; 74941 ; Annecy-le-Vieux Cedex ; France
  3. Centre de Recherches Math茅matiques ; Concordia University ; Montr茅al ; QC ; Canada
  4. SISSA ; via Bonomea ; 265 ; 34136 ; Trieste ; Italia
  
• 关键词：Quantum integrable systems ; Combinatorics ; Mathematical physics ; Symmetric polynomials ; 6 Vertex model
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：41
• 期：3
• 页码：843-866
• 全文大小：414 KB
• 参考文献：1. Behrend, R.E., Knight, V.A.: Higher spin alternating sign matrices. Electron. J. Combin. 14, 38 (2007). Research Paper 83, http://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1r83
  2. Bressoud, D (1999) Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. MAA Spectrum Mathematical Association of America, Washington, DC CrossRef
  3. Caradoc, A., Foda, O., Kitanine, N.: Higher spin vertex models with domain wall boundary conditions. J. Stat. Mech. 2006(03), P03012 (2006). doi:10.1088/1742-5468/2006/03/P03012
  4. de Gier, J.: Loops, matchings and alternating-sign matrices. Discrete Math 298(1鈥?), 365鈥?88 (2005). doi:10.1016/j.disc.2003.11.060
  5. Dow, A., Foda, O.: On the domain wall partition functions of level-1 affine so(n) vertex models. J. Stat. Mech. 2006(05), P05010 (2006). doi:10.1088/1742-5468/2006/05/P05010
  6. Feigin, B., Jimbo, M., Miwa, T., Mukhin, E.: Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. Int. Math. Res. Not. 2003(18), 1015鈥?034 (2003)
  7. Foda, O., Wheeler, M., Zuparic, M.: Domain wall partition functions and KP. J. Stat. Mech. 2009(3), P03017 (2009). doi:10.1088/1742-5468/2009/03/P03017
  8. Fonseca, T.: Alternating sign matrices, completely packed loops and plane partitions, Ph.D. thesis, Universit茅 Pierre et Marie Curie, (2010). http://tel.archives-ouvertes.fr/tel-00521884/fr/
  9. Fulton, W, Harris, J (1991) Representation Theory, Graduate Texts in Mathematics. Springer, New York
  10. Izergin, AG (1987) Partition function of a six-vertex model in a finite volume. Dokl. Akad. Nauk SSSR 297: pp. 331-333
  11. Korepin, V.: Calculation of norms of Bethe wave functions. Comm. Math. Phys. 86(3), 391鈥?18 (1982). http://www.springerlink.com/content/x6j8w6j351673l32
  12. Kulish, P, Reshetikhin, N, Sklyanin, E (1981) Yang-Baxter equation and representation theory: I. Lett. Math. Phys. 5: pp. 393-403 CrossRef
  13. Kuperberg, G.: Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Not. 3, 139鈥?50 (1996). doi:10.1155/S1073792896000128
  14. Kuperberg, G.: Symmetry classes of alternating-sign matrices under one roof. Ann. of Math. 3, 835鈥?66 (2002). doi:10.2307/3597283
  15. Lascoux, A.: Gaudin functions, and Euler-Poincar茅 characteristics (2007). arXiv:0709.1635
  16. Macdonald, IG (1979) Symmetric functions and Hall polynomials. Oxford University Press Inc., Oxford
  17. Mills, W, Robbins, D, Rumsey, H (1983) Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A 34: pp. 340-359 CrossRef
  18. Mills, W, Robbins, D, Rumsey, H (1986) Self-complementary totally symmetric plane partitions. J. Combin. Theory Ser. A 42: pp. 277-292 CrossRef
  19. Okada, S.: Enumeration of symmetry classes of alternating sign matrices and characters of classical groups. J. Algebraic Combin. 23(1), 43鈥?9 (2006)
  20. Pimenta, R.A., Martins, M.J.: The Yang-Baxter equation for PT invariant 19-vertex models. J. Phys. A 44(8), 085205 (2011). arXiv:1010.1274
  21. Reshetikhin, N.: Lectures on the Integrability of the 6-Vertex Model, Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, Lecture Notes of the Les Houches Summer School, vol. 89 (2010). arXiv:1010.5031
  22. Rosengren, H (2009) An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices. Adv. Appl. Math. 43: pp. 137-155 CrossRef
  23. Stroganov, Yu G (2006) Izergin-Korepin determinant at a third root of unity. Theoret. Math. Phys. 146: pp. 53-62 CrossRef
  24. Takasaki, K.: KP and Toda tau functions in Bethe Ansatz, New Trends in Quantum Integrable Systems, pp. 373鈥?91, World Scientific Publishing, Hackensack (2011). arXiv:1003.3071
  25. Yang, W-L, Zhang, Y-Z (2009) Partition function of the eight-vertex model with domain wall boundary condition. J. Math. Phys. 50: pp. 083518,14 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

The determinantal form of the partition function of the \(6\) -vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur polynomial. Caradoc, Foda and Kitanine computed the partition function of the higher spin generalization of the \(6\) -vertex model. In the present work, it is shown that for a special value of the crossing parameter, referred to as the combinatorial point, the partition function reduces to a Macdonald polynomial.