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TMF, 2008, Volume 154, Number 3, Pages 387–408 (Mi tmf6178)  

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

Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices

P. Zinn-Justina, Ph. Di Francescob

a Laboratoire de Physique Théorique et Modèles Statistiques, Univ Paris-Sud, Orsay, France
b Service de Physique Théorique de Saclay, Gif sur Yvette Cedex, France

Abstract: We present multiple-residue integral formulas for partial sums in the basis of link patterns of the polynomial solution of the level-$1$ $U_q(\widehat{\mathfrak{sl}_2})$ quantum Knizhnik–Zamolodchikov equation at arbitrary values of the quantum parameter $q$. These formulas allow rewriting and generalizing a recent conjecture of Di Francesco connecting these sums to generating polynomials for weighted totally symmetric self-complementary plane partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.

Keywords: loop model, combinatorics, quantum integrability

DOI: https://doi.org/10.4213/tmf6178

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English version:
Theoretical and Mathematical Physics, 2008, 154:3, 331–348

Bibliographic databases:


Citation: P. Zinn-Justin, Ph. Di Francesco, “Quantum Knizhnik–Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices”, TMF, 154:3 (2008), 387–408; Theoret. and Math. Phys., 154:3 (2008), 331–348

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v154/i3/p387

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Di Francesco P., Zinn-Justin P., “Quantum Knizhnik–Zamolodchikov equation: reflecting boundary conditions and combinatorics”, Journal of Statistical Mechanics: Theory and Experiment, 2007, December, P12009, 30 pp.  crossref  mathscinet  isi  scopus
    2. Fonseca T., Zinn-Justin P., “On the doubly refined enumeration of alternating sign matrices and totally symmetric self-complementary plane pa”, Electron. J. Combin., 15:1 (2008), 81, 35 pp.  mathscinet  zmath  isi
    3. de Gier J., Ponsaing A., Shigechi K., “The exact finite size ground state of the O($n=1$) loop model with open boundaries”, Journal of Statistical Mechanics: Theory and Experiment, 2009, April, P04010, 26 pp.  crossref  isi  scopus
    4. de Gier J., Pyatov P., Zinn-Justin P., “Punctured plane partitions and the $q$-deformed Knizhnik–Zamolodchikov and Hirota equations”, J. Combin. Theory Ser. A, 116:4 (2009), 772–794  crossref  mathscinet  zmath  isi  elib  scopus
    5. Fonseca T., Zinn-Justin P., “On some ground state components of the O(1) loop model”, Journal of Statistical Mechanics: Theory and Experiment, 2009, March, P03025, 29 pp.  crossref  isi  scopus
    6. Fischer I., “An operator formula for the number of halved monotone triangles with prescribed bottom row”, J. Combin. Theory Ser. A, 116:3 (2009), 515–538  crossref  mathscinet  zmath  isi  elib  scopus
    7. Zinn-Justin P., “A conjectured formula for Fully Packed Loop configurations in a triangle”, Electron. J. Combin., 17:1 (2010), R107, 27 pp.  mathscinet  zmath  isi  elib
    8. 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
    9. Fendley P., Hagendorf Ch., “Ground-state properties of a supersymmetric fermion chain”, J. Stat. Mech. Theory Exp., 2011, P02014  crossref  isi  elib  scopus
    10. Fonseca T., Nadeau Ph., “On some polynomials enumerating Fully Packed Loop configurations”, Adv in Appl Math, 47:3 (2011), 434–462  crossref  mathscinet  zmath  isi  scopus
    11. de Gier J., Lascoux A., Sorrell M., “Deformed Kazhdan-Lusztig elements and Macdonald polynomials”, J Combin Theory Ser A, 119:1 (2012), 183–211  crossref  mathscinet  zmath  isi  scopus
    12. F. Colomo, A. G. Pronko, “An approach for calculating correlation functions in the six-vertex model with domain wall boundary conditions”, Theoret. and Math. Phys., 171:2 (2012), 641–654  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    13. Shigechi K., Zinn-Justin P., “Path Representation of Maximal Parabolic Kazhdan-Lusztig Polynomials”, J. Pure Appl. Algebr., 216:11 (2012), 2533–2548  crossref  mathscinet  zmath  isi  elib  scopus
    14. Behrend R.E., “Multiply-Refined Enumeration of Alternating Sign Matrices”, Adv. Math., 245 (2013), 439–499  crossref  mathscinet  zmath  isi  elib  scopus
    15. Fonseca T., “On Some Polynomials Enumerating Fully Packed Loop Configurations: Evaluation at Negative Values”, Adv. Appl. Math., 51:3 (2013), 446–466  crossref  mathscinet  zmath  isi  elib  scopus
    16. Hagendorf Ch., “Spin Chains with Dynamical Lattice Supersymmetry”, J. Stat. Phys., 150:4 (2013), 609–657  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Motegi K., Sakai K., “K-Theoretic Boson-Fermion Correspondence and Melting Crystals”, J. Phys. A-Math. Theor., 47:44 (2014), 445202  crossref  mathscinet  zmath  adsnasa  isi  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
    19. Di Francesco P., “Integrable Combinatorics”, Xviith International Congress on Mathematical Physics, ed. Jensen A., World Scientific Publ Co Pte Ltd, 2014, 29–51  mathscinet  zmath  isi
    20. 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
    21. Aigner F., “Fully Packed Loop Configurations: Polynomiality and Nested Arches”, Electron. J. Comb., 25:1 (2018), P1.27  mathscinet  zmath  isi
    22. Striker J., “Permutation Totally Symmetric Self-Complementary Plane Partitions”, Ann. Comb., 22:3 (2018), 641–671  crossref  mathscinet  isi  scopus
    23. Paul Zinn-Justin, “Loop Models and $K$-Theory”, SIGMA, 14 (2018), 069, 48 pp.  mathnet  crossref
    24. Knutson A. Zinn-Justin P., “Grassmann-Grassmann Conormal Varieties, Integrability, and Plane Partitions”, Ann. Inst. Fourier, 69:3 (2019), 1087–1145  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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