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 for $n\le 2$ only) functional equations is given. For odd levels the $\widehat{s{l}_2}$ 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 $\widehat{s{l}_n}$ are obtained. Connection between dilogarithm identities and algebraic $K$-theory (torsion in $K_3(\mathbf{R})$) 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). Connection between "finite-dimensional part of crystal base" and Robinson-Schensted-Knuth correspondence is considered.

96 pages

References in corpus (5)

• Quasi-particles models for the representations of Lie algebras and geometry of flag manifold
• Quantum groups and zeta-functions
• Dilogarithm Identities in Conformal Field Theory and Group Homology
• Crystal bases, dilogarithm identities and torsion in algebraic K-groups
• Fusion algebra and Verlinde's formula

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