Dilogarithm Identities

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 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 are obtained. Connection between dilogarithm identities and algebraic K-theory (torsion in K₃(R)) is discussed. Relations between crystal bases, branching functions b_λ^kΛ₀(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_λ^kΛ₀(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.

Citing Article(s) :

1. Journal of the Physical Society of Japan 65 (1996) pp. 1617-1621: Higher-Spin Generalization of Fractional Exclusion Statistics Takahiro Fukui, Norio Kawakami and Sung-Kil Yang

2. Journal of the Physical Society of Japan 67 (1998) pp. 1163-1174: Construction of the Quantum Integrable Equations on Space-Time Lattice Based on the Lattice W Algebra Rei Inoue and Kazuhiro Hikami

3. Journal of the Physical Society of Japan 68 (1999) pp. 55-60: The \rbbZ_N Symmetric Quantum Lattice Field Theory The Quantum Group Symmetry, the Yang–Baxter Equation, and the Integrals of Motion Kazuhiro Hikami

4. Journal of the Physical Society of Japan 68 (1999) pp. 380-385: The Quantum Volterra Model and the Lattice Sine–Gordon System.– Construction of the Baxter Q Operator and the Integrals of Motion Kazuhiro Hikami and Rei Inoue

5. Journal of the Physical Society of Japan 68 (1999) pp. 1843-1846: Quantum Integrable Model on ( 2 + 1)-D Lattice Rei Inoue and Kazuhiro Hikami