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Prog. Theor. Phys. Supplement No.118 (1995) pp. 61-142

[ Full Text PDF : FREE ACCESS (5668K) ]

Dilogarithm Identities

Anatol N. Kirillov

Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153
and
Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011

Abstract:

We study the dilogarithm identities from algebraic, analytic, asymptotic, K-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all!) can be obtained by using the five-term relation only. Among those the Coxeter, Lewin, Loxton and Browkin ones are contained. Accessibility of Lewin's one variable and Ray's multivariable (here from n ≤2 only) functional equations is given. For odd levels and \hatsl2 case of Kuniba-Nakanishi's dilogarithm conjecture is proven and additional results about remainder them are obtained. The connections between dilogarithm identities and Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic behavior are discussed. Some new results about the string functions for level k vacuum representation of the affine Lie algebra \hatsln are obtained. Connection between dilogarithm identities and algebraic K-theory (torsion in K3(R)) is discussed. Relations between crystal bases, branching functions bλkΛ0(q) and Kostka-Foulkes polynomials (Lusztig's q-analog of weight multiplicity) are considered. The Melzer and Milne conjectures are proven. In some special cases we are proving that the branching functions bλkΛ0(q) are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). Connection between “finite-dimensional part of crystal base” and Robinson-Schensted-Knuth correspondence is considered.


URL : http://ptp.ipap.jp/link?PTPS/118/61/
DOI : 10.1143/PTPS.118.61

[ Full Text PDF : FREE ACCESS (5668K) ] Citation:

References:

  1. K. Aomoto, Nagoya Math. J. 88 (1982), 55.
  2. D. Acreman and J. H. Loxton, Aequat. Math. 30 (1986), 106.
  3. G. Andrews, The Theory of Partitions (Addison-Wesley Publishing Company, 1976).
  4. G. Andrews, Pacific J. Math. 114 (1984), 267.
  5. G. Andrews, Regional Conference Series in Mathematics, ISSN 0160-7642 (1985), No. 66.
  6. G. Andrews, R. Baxter, D. Bressoud, W. Burge, P. Forrester and G. Viennot, Europ. J. Combinatorics 8 (1987), 341.
  7. G. Andrews, R. Baxter and P. Forrester, J. Stat. Phys. 35 (1984), 193.
  8. G. Andrews and D. Bressoud, Rocky Mtn. J. Math. 15 (1985), 225.
  9. A. Agarwal, G. Andrews and D. Bressoud, J. Indian Math. Soc. 51 (1987), 57.
  10. W. Bailey, Proc. London Math. Soc. (2), 50 (1949), 1.
  11. V. Bazhanov and N. Reshetikhin, J. of Phys. A 23 (1990), 1477[IoP STACKS].
  12. A. Beilinson, A. Goncharov, V. Schechtman and A. Varchenko, The Grothendieck Festschrift, vol. 1. (Birkhauser, Progress in Math. Series, 86 Boston, 1990), p. 135.
  13. A. Berkovich, Preprint Univ. Bonn (1994), p. 31; hep-th/9403073[e-print arXiv].
  14. S. Bloch, Proc. of the International Symp. on Alg. Geom. (Kyoto Univ., Kyoto, 1977) (Kinokuniya Book Store, 1978), p. 103.
  15. P. Bouwknegt and K. Schoutens, Phys. Rep. 223 (1993), 183[CrossRef].
  16. D. Bressoud, Mem. Amer. Math. Soc. 224 (1980), 54.
  17. D. Bressoud, J. Number Theory 16 (1983), 235.
  18. D. Bressoud, Discrete Math. 73 (1988/89), 37.
  19. D. Bressoud, Lecture Notes in Math. 1395 (1987), 140.
  20. D. Bressoud, in q-series and Partitions, ed. D. Stanton, IMA Volumes in Math. and Applic., v. 18 (Springer-Verlag, New-York, 1989), p. 45.
  21. D. Bressoud, in Ramanujan Revisited, ed. G. Andrews et al. (Academic Press, Inc., 1988), p. 57.
  22. J. Browkin, in K-Theory 3 (1989), p. 29.
  23. J. Browkin, Aequat. Math. 30 (1986), chap. 11 p. 233.
  24. R. K. Brylinski, J. Amer. Math. Soc. 2 (1989), 517.
  25. W. Burger, Europ. J. Comb. 3 (1982), 195.
  26. J. L. Cathelineau, Seminaire BOURBAKI, (1992-93), n0 772, p. 23.
  27. H. S. M. Coxeter, Quart. J. Math. Oxford Ser. 6 (1935), 13.
  28. S. Dasmahapatra, Preprint IC/93/91 (1993), p. 13.
  29. S. Dasmahapatra, City Univ. Preprint (1994), p. 22.
  30. E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Adv. Studies in Pure Math. 16 (1988), 17.
  31. E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Adv. Studies in Pure Math. 19, v. 149.
  32. S. Dasmahapatra, R. Kedem, T. R. Klassen, B. M. McCoy and E. Melzer, Int. J. Mod. Phys. 7 (1993), 3617.
  33. S. Dasmahapatra, R. Kedem, B. M. McCoy and E. Melzer, J. Stat. Phys. 74 (1994), 239.
  34. J. L. Dupont and C.-H. Sah, “Dilogarithm Identities in Conformal Field Theory and Group Homology”, Preprint (1993).
  35. F. Essler, V. E. Korepin and K. Schoutens, J. of Phys. A: Math. Gen. 25 (1992), 4115[IoP STACKS].
  36. L. D. Faddeev and R. M. Kashaev, “Quantum Dilogarithm” (preprint, 1993), p. 8.
  37. L. D. Faddeev and A. Yu. Volkov, Preprint HU-TFT-93-29 (1993).
  38. B. Feigin and E. Frenkel, Advances in Sov. Math. 16 (1993).
  39. B. Feigin and D. Fuchs, in Representations of Infinite-Dimensional Lie Groups and Lie Algebras, ed. A. M. Vershik and D. P. Zhelobenko (Gordon and Breach, 1990), p. 465.
  40. B. Feigin, T. Nakanishi and H. Ooguri, Int. J. Mod. Phys. A 1 (1992), 217.
  41. B. L. Feigin and A. V. Stoyanovsky, “Quasi-Particles Models for the Representations of Lie Algebras and Geometry of Flag Manifold”, (preprint, 1993), p. 35.
  42. S. Fishel, “The Combinatorics of Certain Entries in the Two-Parameter Kostka Matrix”, (preprint, 1993), p. 9.
  43. D. Foata, in Higher Combinatorics, ed. M. Aigner (Berlin, 1976), p. 27, (Amsterdam, D. Reidel, 1977) (Proc. NATO Adv. Study Inst.)
  44. O. Foda and Y.-H. Quano, Univ. Melbourne Preprint No. 26 (1994), p. 18.
  45. P. Forrester and R. Baxter, J. Stat. Phys. 38 (1985), 435.
  46. E. Frenkel and A. Szenes, Duke Math. J., Int. Math. Res. Notices 2 (1993), 53.
  47. E. Frenkel and A. Szenes, “Crystal Bases, Dilogarithm Identities and Torsion in Algebraic K-Groups”, (preprint, 1993), p. 33.
  48. A. Garsia, Discrete Math. 99 (1992), 247.
  49. A. Garsia and S. Milne, J. Comb. Theory, Ser. A 31 (1981), 289.
  50. G. Gasper and M. Rahman, Basic Hypergeometric Series (Cambridge Univ. Press, 1990).
  51. I. Gessel and D. Stanton, Trans. Amer. Math. Soc. 277 (1983), 173.
  52. J. Grene, Discrete Math. 68 (1988), 15.
  53. P. Goddard, A. Kent and D. Olive, Phys. Lett. B 152 (1985), 88.
  54. F. Goodman and H. Wenzl, Adv. Math. 82 (1990), 244.
  55. A. Goncharov, Preprint MSRI-058-93 (1993), p. 35.
  56. G.-N. Han, C. R. Acad. Sci. Paris. 314 (1992), Serie I, 969.
  57. U. Jannsen, S. Kleiman and J.-P. Serre (ed)., Proc. Symp. in Pure Math. (AMS, Providence, 1994), v. 55, part 2.
  58. M. Jimbo, K. C. Misra, T. Miwa and M. Okabo, Commun. Math. Phys. 136 (1991), 543.
  59. V. G. Kac, Infinite Dimensional Lie Algebras (Cambridge Univ. Press, 1990).
  60. V. Kac and D. Peterson, Adv. Math. 53 (1984), 125.
  61. V. Kac and M. Wakimoto, Proc. Natl. Acad. Sci. USA 85 (1988), 4956.
  62. V. Kac and M. Wakimoto, Adv. Math. 70 (1988), 156.
  63. V. Kac and M. Wakimoto, “Integrable Highest Modules over Affine Superalgebras and Number Theory”, (preprint, 1994), p. 44.
  64. S.-J. Kang, M. Kashiwara, K. C. Misra, T. Nakashima and A. Nakayashiki, Int. J. Mod. Phys. A 7 (1992), 449.
  65. M. Kashiwara, Duke Math. J. 63 (1991), 465.
  66. R. Kaufmann, Preprint Bonn-TH-94-05, 1994, 19P; hep-th/9405041[e-print arXiv].
  67. R. Kedem, T. R. Klassen, B. M. McCoy and E. Melzer, Phys. Lett. B 304 (1993), 263[CrossRef].
  68. R. Kedem, T. R. Klassen, B. M. McCoy and E. Melzer, Phys. Lett. B 307 (1993), 68[CrossRef].
  69. R. Kedem and B. M. McCoy, J. Stat. Phys. 71 (1993), 865.
  70. R. Kedem, B. M. McCoy and E. Melzer, Preprint ITP-SB-93-19 (1993), p. 27.
  71. A. N. Kirillov, Zap. Nauch. Sem. LOMI (in Russian) 131 (1984), 88.
  72. A. N. Kirillov, Zap. Nauch. Sem. LOMI (in Russian) 134 (1984), 169.
  73. A. N. Kirillov, J. Geom. and Phys. 5 (1988), 365.
  74. A. N. Kirillov, Zap. Nauch. Sem. LOMI (in Russian) 164 (1987), 121; J. Sov. Math. 47 (1989), 2450.
  75. A. N. Kirillov, C. R. Acad. Sci. Paris 315 (1992), Serie I, 497.
  76. A. N. Kirillov, “Fusion Algebra and Verlinde's Formula” (preprint, 1992), p. 9.
  77. A. N. Kirillov, submitted to Algebra and Analysis.
  78. A. N. Kirillov, Preprint LITP-93-22, (1993), p. 10.
  79. A. N. Kirillov, Int. J. Mod. Phys. 7 (1992), 545.
  80. A. N. Kirillov, “Quantum Polylogarithms”, preprint (1994).
  81. A. N. Kirillov and S. V. Kerov, submitted to Algebra and Analysis.
  82. A. N. Kirillov and N. A. Liskova, Preprint UTMS-94-47, Tokyo Univ. (1994), p. 20.
  83. T. Klassen and E. Melzer, Nucl. Phys. B 338 (1990), 485[CrossRef].
  84. A. N. Kirillov and N. Yu. Reshetikhin, J. Sov. Math. 35 (1986), 2627.
  85. A. N. Kirillov and N. Yu. Reshetikhin, J. Sov. Math. 41 (1988), 925.
  86. A. Kuniba, Nucl. Phys. B 389 (1993), 209[CrossRef].
  87. A. Kuniba and T. Nakanishi, Mod. Phys. Lett. A 7 (1992), 3487.
  88. A. Kuniba, T. Nakanishi and J. Suzuki, Mod. Phys. Lett. A 7 (1993), 1649.
  89. A. Lascoux and M.-P. Schützenberger, C. R. Acad. Sci. Paris A 286 (1978), 323.
  90. J. Lepowsky and M. Primc, Contemporary Math. (AMS, Providence, 1985), v. 46.
  91. M. Levine, Ann. Ec. Norm. Sup. 22 (1989), 255.
  92. L. Lewin, Dilogarithm and Associated Functions (North Holland, Amsterdam, 1981).
  93. L. Lewin, J. Austral. Math. Soc. Series A 33 (1982), 302.
  94. L. Lewin, J. Number Theory 19 (1984), 345.
  95. L. Lewin, Aequat. Math. 30 (1986), 1.
  96. L. Lewin (ed). Math. Surveys and Monographs (1991), v. 37.
  97. S. Lichtenbaum, Contemporary Math. 83 (1989), 151.
  98. X.-S. Lin, Preprint IAS (1994), p. 10.
  99. D. E. Littlewood, The Theory of Group Characters (Oxford University Press, 1950).
  100. J. H. Loxton, Acta Mathematica 43 (1984), 155.
  101. J. H. Loxton, Aequat. Math. 30 (1986), chap. 13, p. 287.
  102. G. Lusztig, Adv. Math. 42 (1981), 169.
  103. G. Lusztig, Astérisque 101-102 (1983), 208.
  104. G. Lusztig, Contemporary Math. 82 (1989), 59.
  105. I. G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford University Press, 1979).
  106. E. Melzer, Lett. Math. Phys. 31 (1994), 233.
  107. E. Melzer, Int. J. Mod. Phys. A 9 (1994), 1115.
  108. S. Milne, Discrete Math. 99 (1992), 199.
  109. S. Milne, SIAM J. Math. Anal. 25 (1994), 571.
  110. S. Milne, Proc. Sympos. Pure Math. 49 (1989), (part 2), 323.
  111. S. Milne and G. Lilly, “Consequences of the Al and Gl Bailey transform and Bailey Lemma”, preprint (1992), p. 28.
  112. J. Milnor, Annals of Math. Studies 72 (Princeton Univ. Press, 1971).
  113. J. Milnor, Hyperbolic Geometry: The first 150 years (Bull A. M. S., 1982), v. 6, p. 9.
  114. J. Milnor, Enseign. Math. 29 (1983), 281.
  115. K. C. Misra and T. Miwa, Commun. Math. Phys. 134 (1990), 79.
  116. A. S. Merkuriev and A. A. Suslin, Math. USSR Izv. 36 (1991), 541.
  117. D. Moak, Rocky Mtn. J. Math. 14 (1984), 403.
  118. W. Nahm, A. Recknagel and M. Terhoeven, Mod. Phys. Lett. A 8 (1993), 1835.
  119. J. Oesterlé, Polylogarithmes (Seminarire BOURBAKI, 1992-93), n0762, p. 16.
  120. P. Paule, J. Math. Analysis and Appl. 107 (1985), 255.
  121. P. Paule, The Concept of Bailey Chains (Publ. I. R. M. A., Strasbourg, 1988), Actes 18e Séminaire Lotharingien, p. 53.
  122. Y.-H. Quano, Univ. Melbourne Preprint No. 25 (1994), p. 25.
  123. Y.-H. Quano, Univ. Melbourne Preprint No. 27 (1994), p. 9.
  124. G. Ray, Aequat. Math. 30 (1986), chap. 7, p. 123.
  125. N. Yu. Reshetikhin, Preprint/hep-th/9403105 (1994), p. 17.
  126. B. Richmond and G. Szekeres, J. Austral. Math. Soc. A 31 (1981), 362.
  127. A. Rocha-Caridi, in Vertex Operator in Mathematical Physics, ed. J. Lepowsky, S. Mandelstam and I. M. Singer (Publications of MSRI, Springer-Verlag, 1984), v. 3, p. 451.
  128. L. Rogers, Proc. London Math. Soc. 4 (1907), 169.
  129. M. Rosso, Commun. Math. Phys. 117 (1988), 581[CrossRef].
  130. B. Sagan, The Symmetric Group (Wadsworth & Brooks/Cole, Pacific Grove, CA, 1991).
  131. A. Sergeev, Math. USSR, Sbornik 51 (1985), 419.
  132. V. Serganova, Yale Univ. Preprint (1992), p. 8.
  133. L. J. Slater, Proc. London Math. Soc. 54 (1951), 147.
  134. R. P. Stanley, in Surveys in Combinatorics, ed. E. K. Lloyd (London Math. Soc., Lecture Note Series, No. 82) (Cambridge University Press, Cambridge, 1983), p. 187.
  135. R. P. Stanley, Linear and Multil. Algebra 16 (1984), 3.
  136. J. R. Stembridge, Trans Amer. Math. Soc. 319 (1990), 469.
  137. A. A. Suslin, Proc. of the Steklov Inst. of Math. 4 (1991), 217.
  138. I. Terada, “A Generalization of the Length Symmetry and Variety of N-Stabel Flags”, preprint (1993), p. 26.
  139. M. Terhoeven, Preprint Bonn-HE-92-36 (1992); hep-th/9211120[e-print arXiv].
  140. M. Terhoeven, Preprint Bonn-HE-93-24 (1993), p. 11.
  141. V. Tarasov and A. Varchenko, Preprint UTMS-94-46 (Tokyo Univ., 1994).
  142. K. Ueno and M. Nishizawa, “Quantum Groups and Zeta-Functions”, preprint (1994), p. 12.
  143. A. Varchenko, Multidimensional Hypergeometric Functions in Conformal Field Theory, Algebraic K-Theory, Algebraic Geometry (ICM, Kyoto 1990), v. 1, p. 281.
  144. G. N. Watson, J. London Math. Soc. 4 (1929), 4.
  145. G. N. Watson, Quart. J. Math. Oxford, Ser. 8 (1937), 39.
  146. D. Zagier, Invent. Math. 83 (1986), 285.
  147. D. Zagier, J. Math. Phys. Sci. 22 (1988), 131[AIP Scitation].
  148. D. Zagier, Papers presented at the Ramanujan colloquium, Bombay, 1988, TIFR.
  149. D. Zagier, in Arithmetic Algebraic Geometry, Birkhauser, ed. G. van der Geer e.a. (1991), v. 89, p. 391.
  150. Al. Zamolodchikov, Nucl. Phys. B 342 (1990), 695[CrossRef].
  151. D. Zeiberger and D. Bressoud, Discrete Math. 54 (1985), 201.

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