Dilogarithm identities

TL;DR

This paper explores dilogarithm identities from multiple mathematical perspectives, proving many can be derived from the five-term relation, and establishes connections with algebraic K-theory, partition identities, and representation theory.

Contribution

It demonstrates that numerous dilogarithm identities can be obtained solely from the five-term relation and proves special cases of conjectures related to affine Lie algebras and K-theory.

Findings

Proved the $ ext{Kuniba-Nakanishi}$ dilogarithm conjecture for odd levels.

Established connections between dilogarithm identities and Rogers-Ramanujan-type partition identities.

Proven Melzer and Milne conjectures regarding crystal bases and Kostka-Foulkes polynomials.

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 $\hat{sl_2}$ 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 $\hat{sl_n}$ are obtained. Connection between dilogarithm identities and algebraic $K$-theory (torsion in $K_3({\bf R})$) 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.