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An A$_2$ Bailey lemma
and Rogers-Ramanujan-type identities


Authors: George E. Andrews, Anne Schilling and S. Ole Warnaar
Journal: J. Amer. Math. Soc. 12 (1999), 677-702
MSC (1991): Primary 05A30, 05A19; Secondary 33D90, 33D15, 11P82
DOI: https://doi.org/10.1090/S0894-0347-99-00297-0
Published electronically: April 23, 1999
MathSciNet review: 1669957
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Abstract | References | Similar Articles | Additional Information

Abstract: Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A$_2$ version of the classical Bailey lemma. We apply our result, which is distinct from the A$_2$ Bailey lemma of Milne and Lilly, to derive Rogers-Ramanujan-type identities for characters of the W$_3$ algebra.


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  • 1. A. K. Agarwal, G. E. Andrews and D. M. Bressoud, The Bailey lattice, J. Ind. Math. Soc. 51 (1987), 57-73. MR 90i:11113
  • 2. G. E. Andrews, An analytic generalization of the Rogers-Ramanujan identities for odd moduli, Proc. Nat. Acad. Sci. USA 71 (1974), 4082-4085. MR 50:4473
  • 3. G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2 (Addison-Wesley, Reading, Massachusetts, 1976). MR 58:27738
  • 4. G. E. Andrews, Multiple series Rogers-Ramanujan type identities, Pacific J. Math. 114 (1984), 267-283. MR 86c:11084
  • 5. G. E. Andrews, R. J. Baxter, D. M. Bressoud, W. H. Burge, P. J. Forrester and G. Viennot, Partitions with prescribed hook differences, Europ. J. Combinatorics 8 (1987), 341-350. MR 89g:05014
  • 6. W. N. Bailey, Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949), 1-10. MR 9:585b
  • 7. D. M. Bressoud, A generalization of the Rogers-Ramanujan identities for all moduli, J. Combin. Theory Ser. A 27 (1979), 64-68. MR 81c:05008
  • 8. D. M. Bressoud, An analytic generalization of the Rogers-Ramanujan identities with interpretation, Quart. J. Math. Oxford (2) 31 (1980), 385-399. MR 83d:05011
  • 9. D. M. Bressoud, Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Memoirs Amer. Math. Soc. 24 (1980), 1-54. MR 81i:10019
  • 10. D. M. Bressoud, The Bailey lattice: An introduction, in Ramanujan Revisited, pp. 57-67, G. E. Andrews et al. eds. (Academic Press, New York, 1988). MR 89f:05018
  • 11. F. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 140-156. MR 58:25442
  • 12. V. A. Fateev and S. L. Lykyanov, The models of two-dimensional conformal quantum field theory with $Z_n$ symmetry, Int. J. Mod. Phys. A 3 (1988), 507-520. MR 89b:81133
  • 13. W. Fulton, Young tableaux: with applications to representation theory and geometry, London Math. Soc. student texts 35, Cambridge University Press (1997). CMP 97:16
  • 14. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Vol. 35 (Cambridge University Press, Cambridge, 1990). MR 91d:33034
  • 15. I. M. Gessel and C. Krattenthaler, Cylindric partitions, Trans. Amer. Math. Soc. 349 (1997), 429-479. MR 97e:05014
  • 16. B. Gordon, A combinatorial generalization of the Rogers-Ramanujan identities, Amer. J. Math. 83 (1961), 393-399. MR 23:A809
  • 17. G. H. Hardy, Ramanujan (Cambridge University Press, London and New York, 1940). MR 3:71d
  • 18. G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Character formulae of $\widehat{sl}_n$-modules and inhomogeneous paths, Nucl. Phys. B 536 [PM] (1999), 575-616. CMP 99:06
  • 19. M. Jimbo, T. Miwa and M. Okado, Local state probabilities of solvable lattice models: An A$_{n-1}^{(1)}$ family, Nucl. Phys. B 300 [FS22] (1988), 74-108. MR 90d:82039
  • 20. A. N. Kirillov, Dilogarithm identities, Prog. Theor. Phys. Suppl. 118 (1995), 61-142. MR 96h:11102
  • 21. A. N. Kirillov, New combinatorial formula for Hall-Littlewood polynomials, math.QA/9803006.
  • 22. A. N. Kirillov and N. Yu. Reshetikhin, The Bethe Ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988), 925-955.
  • 23. C. Krattenthaler, Generating functions for plane partitions of a given shape, Manuscripta Math. 69 (1990), 173-202. MR 92a:05010
  • 24. A. Kuniba, T. Nakanishi and J. Suzuki, Ferro- and antiferro-magnetizations in RSOS models, Nucl. Phys. B 356 (1991), 750-774. MR 92m:82043
  • 25. I. G. Macdonald, Affine root systems and Dedekind's $\eta$-function, Inv. Math. 15 (1972), 91-143. MR 50:9996
  • 26. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition (Clarendon Press, Oxford, 1995). MR 96h:05207
  • 27. P. A. MacMahon, Combinatory Analysis, Vol. 2 (Cambridge University Press, London and New York, 1916).
  • 28. S. C. Milne, Classical partition functions and the $U(n+1)$ Rogers-Selberg identity, Discrete Math. 99 (1992), 199-246. MR 93m:33017
  • 29. S. C. Milne, A $q$-analog of a Whipple's transformation for hypergeometric series in $U(n)$, Adv. in Math. 108 (1994), 1-76. MR 96h:33011
  • 30. S. C. Milne and G. M. Lilly, The $A_{\ell}$ and $C_{\ell}$ Bailey transform and lemma, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 258-263. MR 93g:33016
  • 31. S. C. Milne and G. M. Lilly, Consequences of the $A_{\ell}$ and $C_{\ell}$ Bailey transform and Bailey lemma, Discrete Math. 139 (1995), 319-346. MR 96g:33025
  • 32. S. Mizoguchi, The structure of representations for the W$_3$ minimal model, Int. J. Mod. Phys. A 6 (1991), 133-162. MR 92k:81056
  • 33. T. Nakanishi, Non-unitary minimal models and RSOS models, Nucl. Phys. B 334 (1990), 745-766.
  • 34. P. Paule, On identities of the Rogers-Ramanujan type, J. Math. Anal. Appl. 107 (1985), 225-284. MR 86i:11055
  • 35. L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
  • 36. L. J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (2) 16 (1917), 315-336.
  • 37. L. J. Rogers and S. Ramanujan, Proof of certain identities in combinatory analysis, Proc. Cambridge Phil. Soc. 19 (1919), 211-216.
  • 38. A. Schilling and S. O. Warnaar, Inhomogeneous lattice paths, generalized Kostka polynomials and A$_{n-1}$ supernomials, math.QA/9802111. To appear in Commun. Math. Phys.
  • 39. I. J. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. (1917), 302-321.
  • 40. A. B. Zamolodchikov, Infinite additional symmetries in two-dimensional conformal quantum field theory, Teo. Mat. Fiz. 65 (1985), 347-359.

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Additional Information

George E. Andrews
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: andrews@math.psu.edu

Anne Schilling
Affiliation: Instituut voor Theoretische Fysica, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Email: schillin@wins.uva.nl

S. Ole Warnaar
Email: warnaar@wins.uva.nl

DOI: https://doi.org/10.1090/S0894-0347-99-00297-0
Keywords: A$_2$ Bailey lemma, Rogers--Ramanujan identities
Received by editor(s): August 8, 1998
Published electronically: April 23, 1999
Additional Notes: The second author was supported by the “Stichting Fundamenteel Onderzoek der Materie”.
The third author was supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
Article copyright: © Copyright 1999 American Mathematical Society

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