Dilogarithm Identities
Abstract
We study the dilogarithm identities from algebraic, analytic, asymptotic, Ktheoretic, combinatorial and representationtheoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all!) can be obtained by using the fiveterm 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 hat{sl_2} case of KunibaNakanishi's dilogarithm conjecture is proven and additional results about remainder them are obtained. The connections between dilogarithm identities and RogersRamanujanAndrewsGordon 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 Ktheory (torsion in K_3(R)) is discussed. Relations between crystal bases, branching functions b_λ^{kΛ_0}(q) and KostkaFoulkes polynomials (Lusztig's qanalog 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^{kΛ_0}(q) are equal to an appropriate limit of Kostka polynomials (the socalled Thermodynamic Bethe Ansatz limit). Connection between ``finitedimensional part of crystal base'' and RobinsonSchenstedKnuth correspondence is considered.
 Publication:

Progress of Theoretical Physics Supplement
 Pub Date:
 1995
 DOI:
 10.1143/PTPS.118.61
 arXiv:
 arXiv:hepth/9408113
 Bibcode:
 1995PThPS.118...61K
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Quantum Algebra
 EPrint:
 96 pages