Geodesic
Matrix Bailey Lemma and the Star-Triangle Relation
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.
Keywords: elliptic hypergeometric functions; Bailey lemma; star-triangle relation. 
@article{SIGMA_2018_14_a120,
     author = {Kamil Yu. Magadov and Vyacheslav P. Spiridonov},
     title = {Matrix {Bailey} {Lemma} and the {Star-Triangle} {Relation}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a120/}
}
TY  - JOUR
AU  - Kamil Yu. Magadov
AU  - Vyacheslav P. Spiridonov
TI  - Matrix Bailey Lemma and the Star-Triangle Relation
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a120/
LA  - en
ID  - SIGMA_2018_14_a120
ER  - 
%0 Journal Article
%A Kamil Yu. Magadov
%A Vyacheslav P. Spiridonov
%T Matrix Bailey Lemma and the Star-Triangle Relation
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a120/
%G en
%F SIGMA_2018_14_a120
Kamil Yu. Magadov; Vyacheslav P. Spiridonov. Matrix Bailey Lemma and the Star-Triangle Relation. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a120/

[1] Andrews G. E., “Multiple series Rogers–Ramanujan type identities”, Pacific J. Math., 114 (1984), 267–283 | DOI | MR | Zbl

[2] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[3] Bailey W. N., “Identities of the Rogers–Ramanujan type”, Proc. London Math. Soc., 50 (1948), 1–10 | DOI | MR | Zbl

[4] Baxter R. J., “Hard hexagons: exact solution”, J. Phys. A: Math. Gen., 13 (1980), L61–L70 | DOI | MR

[5] Baxter R. J., Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982 | MR | Zbl

[6] Bazhanov V. V., Kels A. P., Sergeev S. M., “Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs”, J. Phys. A: Math. Theor., 49 (2016), 464001, 44 pp., arXiv: 1602.07076 | DOI | MR | Zbl

[7] Bressoud D. M., “A matrix inverse”, Proc. Amer. Math. Soc., 88 (1983), 446–448 | DOI | MR | Zbl

[8] Derkachov S. E., “Factorization of the $R$-matrix. I”, J. Math. Sci., 143 (2007), 2773–2790, arXiv: math.QA/0503396 | DOI | MR

[9] Derkachov S. E., Karakhanyan D., Kirschner R., “Yang–Baxter ${\mathcal R}$-operators and parameter permutations”, Nuclear Phys. B, 785 (2007), 263–285, arXiv: hep-th/0703076 | DOI | MR | Zbl

[10] Derkachov S. E., Spiridonov V. P., “Yang–Baxter equation, parameter permutations, and the elliptic beta integral”, Russian Math. Surveys, 68 (2013), 1027–1072, arXiv: 1205.3520 | DOI | MR

[11] Derkachov S. E., Spiridonov V. P., “Finite-dimensional representations of the elliptic modular double”, Theoret. and Math. Phys., 183 (2015), 597–618, arXiv: 1310.7570 | DOI | MR | Zbl

[12] Frenkel I. B., Turaev V. G., “Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions”, The Arnold–Gelfand Mathematical Seminars, Birkhäuser Boston, Boston, MA, 1997, 171–204 | DOI | MR | Zbl

[13] Paule P., “On identities of the Rogers–Ramanujan type”, J. Math. Anal. Appl., 107 (1985), 255–284 | DOI | MR | Zbl

[14] Rahman M., “An integral representation of a $_{10}\varphi_9$ and continuous bi-orthogonal $_{10}\varphi_9$ rational functions”, Canad. J. Math., 38 (1986), 605–618 | DOI | MR | Zbl

[15] Rains E. M., “Multivariate quadratic transformations and the interpolation kernel”, SIGMA, 14 (2018), 019, 69 pp., arXiv: 1408.0305 | DOI | MR | Zbl

[16] Rastelli L., Razamat S. S., “The supersymmetric index in four dimensions”, J. Phys. A: Math. Theor., 50 (2017), 443013, 34 pp., arXiv: 1608.02965 | DOI | MR | Zbl

[17] Spiridonov V. P., “On the elliptic beta function”, Russian Math. Surveys, 56 (2001), 185–186 | DOI | MR | Zbl

[18] Spiridonov V. P., “An elliptic incarnation of the Bailey chain”, Int. Math. Res. Not., 2002 (2002), 1945–1977 | DOI | MR | Zbl

[19] Spiridonov V. P., “A Bailey tree for integrals”, Theoret. and Math. Phys., 139 (2004), 536–541, arXiv: math.CA/0312502 | DOI | MR | Zbl

[20] Spiridonov V. P., “Essays on the theory of elliptic hypergeometric functions”, Russian Math. Surveys, 63 (2008), 405–472, arXiv: 0805.3135 | DOI | MR | Zbl

[21] Spiridonov V. P., Warnaar S. O., “Inversions of integral operators and elliptic beta integrals on root systems”, Adv. Math., 207 (2006), 91–132, arXiv: math.CA/0411044 | DOI | MR | Zbl

[22] Takhtadzhan L. A., Faddeev L. D., “The quantum method of inverse problem and the Heisenberg XYZ model”, Russian Math. Surveys, 34:5 (1979), 11–68 | DOI | MR

[23] Warnaar S. O., “50 years of Bailey's lemma”, Algebraic Combinatorics and Applications (Gößweinstein, 1999), Springer, Berlin, 2001, 333–347, arXiv: 0910.2062 | DOI | MR | Zbl

[24] Warnaar S. O., “Extensions of the well-poised and elliptic well-poised Bailey lemma”, Indag. Math. (N.S.), 14 (2003), 571–588, arXiv: math.CA/0309241 | DOI | MR | Zbl

[25] Zudilin W., Hypergeometric heritage of W.N. Bailey. With an appendix: Bailey's letters to F Dyson, arXiv: 1611.08806