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TMF, 2006, Volume 146, Number 1, Pages 65–76 (Mi tmf2009)  

This article is cited in 24 scientific papers (total in 24 papers)

Izergin–Korepin Determinant at a Third Root of Unity

Yu. G. Stroganov

Institute for High Energy Physics

Abstract: We consider the partition function of the inhomogeneous six-vertex model defined on an $(n\times n)$ square lattice. This function depends on $2n$ spectral parameters $x_i$ and $y_i$ attached to the respective horizontal and vertical lines. In the case of the domain-wall boundary conditions, it is given by the Izergin–Korepin determinant. For $q$ being an $N$-th root of unity, the partition function satisfies a special linear functional equation. This equation is particularly simple and useful when the crossing parameter is $\eta=2\pi/3$, i. e., $N = 3$. It is well known, for example, that the partition function is symmetric in both the $\{x\}$ and the $\{y\}$ variables. Using the abovementioned equation, we find that in the case of $\eta=2\pi/3$, it is symmetric in the union $\{x\}\cup\{y\}$. In addition, this equation can be used to solve some of the problems related to enumerating alternating-sign matrices. In particular, we reproduce the refined alternating-sign matrix enumeration discovered by Mills, Robbins, and Rumsey and proved by Zeilberger, and we obtain formulas for the doubly refined enumeration of these matrices.

Keywords: alternating-sign matrices, enumeration, square-ice model

DOI: https://doi.org/10.4213/tmf2009

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English version:
Theoretical and Mathematical Physics, 2006, 146:1, 53–62

Bibliographic databases:


Citation: Yu. G. Stroganov, “Izergin–Korepin Determinant at a Third Root of Unity”, TMF, 146:1 (2006), 65–76; Theoret. and Math. Phys., 146:1 (2006), 53–62

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Razumov, Yu. G. Stroganov, “Enumerations of half-turn-symmetric alternating-sign matrices of odd order”, Theoret. and Math. Phys., 148:3 (2006), 1174–1198  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. A. V. Razumov, Yu. G. Stroganov, “Enumeration of quarter-turn-symmetric alternating-sign matrices of odd order”, Theoret. and Math. Phys., 149:3 (2006), 1639–1650  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Razumov, AV, “Bethe roots and refined enumeration of alternating-sign matrices”, Journal of Statistical Mechanics-Theory and Experiment, 2006, P07004  crossref  mathscinet  isi  scopus
    4. Fonseca T, Zinn-Justin P, “On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane partitions”, Electronic Journal of Combinatorics, 15:1 (2008), 81  mathscinet  zmath  isi
    5. A. V. Razumov, Yu. G. Stroganov, “Three-coloring statistical model with domain wall boundary conditions: Functional equations”, Theoret. and Math. Phys., 161:1 (2009), 1325–1339  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. A. V. Razumov, Yu. G. Stroganov, “Three-coloring statistical model with domain wall boundary conditions: Trigonometric limit”, Theoret. and Math. Phys., 161:2 (2009), 1451–1459  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. J.-Ch. Aval, “The symmetry of the partition function of some square ice models”, Theoret. and Math. Phys., 161:3 (2009), 1582–1589  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. Fischer, I, “More refined enumerations of alternating sign matrices”, Advances in Mathematics, 222:6 (2009), 2004  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Galleas W., “Functional relations for the six-vertex model with domain wall boundary conditions”, J. Stat. Mech. Theory Exp., 2010, P06008  crossref  isi  elib  scopus  scopus
    10. Karklinsky M., Romik D., “A formula for a doubly refined enumeration of alternating sign matrices”, Adv. in Appl. Math., 45:1 (2010), 28–35  crossref  mathscinet  zmath  isi  scopus  scopus
    11. Rosengren H., “The three-colour model with domain wall boundary conditions”, Adv in Appl Math, 46:1–4 (2011), 481–535  crossref  mathscinet  zmath  isi  scopus  scopus
    12. Cantini L., Sportiello A., “Proof of the Razumov-Stroganov conjecture”, J Combin Theory Ser A, 118:5 (2011), 1549–1574  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. Galleas W., “A new representation for the partition function of the six-vertex model with domain wall boundaries”, J. Stat. Mech. Theory Exp., 2011, no. 1, P01013, 12 pp.  crossref  mathscinet  isi  elib  scopus  scopus
    14. Brubaker B., Bump D., Friedberg S., “Schur Polynomials and The Yang–Baxter Equation”, Comm Math Phys, 308:2 (2011), 281–301  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    15. Behrend R.E. Di Francesco Ph. Zinn-Justin P., “A Doubly-Refined Enumeration of Alternating Sign Matrices and Descending Plane Partitions”, J. Comb. Theory Ser. A, 120:2 (2013), 409–432  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. Behrend R.E., “Multiply-Refined Enumeration of Alternating Sign Matrices”, Adv. Math., 245 (2013), 439–499  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    17. Ayyer A. Romik D., “New Enumeration Formulas for Alternating Sign Matrices and Square Ice Partition Functions”, Adv. Math., 235 (2013), 161–186  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    18. Romik D., “Connectivity Patterns in Loop Percolation i: the Rationality Phenomenon and Constant Term Identities”, Commun. Math. Phys., 330:2 (2014), 499–538  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    19. Rosengren H., “Special Polynomials Related To the Supersymmetric Eight-Vertex Model: a Summary”, Commun. Math. Phys., 340:3 (2015), 1143–1170  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    20. Gorin V. Panova G., “Asymptotics of Symmetric Polynomials With Applications To Statistical Mechanics and Representation Theory”, Ann. Probab., 43:6 (2015), 3052–3132  crossref  mathscinet  zmath  isi  scopus  scopus
    21. Rosengren H., “Elliptic Pfaffians and solvable lattice models”, J. Stat. Mech.-Theory Exp., 2016, 083106  crossref  mathscinet  isi  elib  scopus
    22. de Gier J. Jacobsen J.L. Ponsaing A., “Finite-Size Corrections For Universal Boundary Entropy in Bond Percolation”, SciPost Phys., 1:2 (2016), UNSP 012  crossref  isi
    23. Behrend R.E. Fischer I. Konvalinka M., “Diagonally and Antidiagonally Symmetric Alternating Sign Matrices of Odd Order”, Adv. Math., 315 (2017), 324–365  crossref  mathscinet  zmath  isi  scopus  scopus
    24. Ayyer A. Behrend R.E., “Factorization Theorems For Classical Group Characters, With Applications to Alternating Sign Matrices and Plane Partitions”, J. Comb. Theory Ser. A, 165 (2019), 78–105  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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