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**Dilogarithm identities.**
*(English)*
Zbl 0894.11052

Inami, Takeo (ed.) et al., Quantum field theory, integrable models and beyond. Proceedings of a workshop, Kyoto, Japan, February 14-17, 1994. Kyoto: Kyoto University, Yukawa Inst. for Theoretical Physics, Prog. Theor. Phys., Suppl. 118, 61-142 (1995).

The main purpose of this paper, based on a series of lectures given at Japanese Universities, is a survey of dilogarithm identities and related topics. It contains a wealth of information and an extensive list of references. The contents are conveniently described in the author’s summary.

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 for \(n\leq 2\) only) functional equations is given. For odd levels the \(\widehat{sl}_2\) case of Kuniba-Nakanishi’s dilogarithm conjecture is proven and additional results about remainder term 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 \(\widehat{sl}_n\) are obtained. Connection between dilogarithm identities and algebraic \(K\)-theory (torsion in \(K_3(\mathbb R))\) is discussed. Relations between crystal basis, branching functions \(b_\lambda^{k\Lambda_0} (q)\) 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 \(b_\lambda^{k\Lambda_0} (q)\) are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). The connection between the “finite-dimensional part of crystal base” and the Robinson-Schensted-Knuth correspondence is considered.

For the entire collection see [Zbl 0839.00030].

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 for \(n\leq 2\) only) functional equations is given. For odd levels the \(\widehat{sl}_2\) case of Kuniba-Nakanishi’s dilogarithm conjecture is proven and additional results about remainder term 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 \(\widehat{sl}_n\) are obtained. Connection between dilogarithm identities and algebraic \(K\)-theory (torsion in \(K_3(\mathbb R))\) is discussed. Relations between crystal basis, branching functions \(b_\lambda^{k\Lambda_0} (q)\) 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 \(b_\lambda^{k\Lambda_0} (q)\) are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). The connection between the “finite-dimensional part of crystal base” and the Robinson-Schensted-Knuth correspondence is considered.

For the entire collection see [Zbl 0839.00030].

Reviewer: O.Ninnemann (Berlin)

### MSC:

11Z05 | Miscellaneous applications of number theory |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

33-02 | Research exposition (monographs, survey articles) pertaining to special functions |

11G55 | Polylogarithms and relations with \(K\)-theory |

33B15 | Gamma, beta and polygamma functions |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

17B68 | Virasoro and related algebras |

11P81 | Elementary theory of partitions |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

19M05 | Miscellaneous applications of \(K\)-theory |

### Keywords:

thermodynamic Bethe Ansatz limit; Virasoro algebras; Kac-Moody algebras; conformal field theory; survey; dilogarithm identities; functional equations; Kuniba-Nakanishi’s dilogarithm; partition identities; string functions; affine Lie algebra; algebraic \(K\)-theory; crystal basis; branching functions; Kostka-Foulkes polynomials; Melzer and Milne conjectures; Robinson-Schensted-Knuth correspondence
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\textit{A. N. Kirillov}, Prog. Theor. Phys., Suppl. 118, 61--142 (1995; Zbl 0894.11052)

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### Digital Library of Mathematical Functions:

§25.12(i) Dilogarithms ‣ §25.12 Polylogarithms ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions### Online Encyclopedia of Integer Sequences:

a(n) = numerator(n!/n^2).Decimal expansion of the dilogarithm of (the golden mean minus 1), Li_2(phi-1).