Dilogarithm Identities
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概要
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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-__<2 only) functional equations is given. For odd levels the [numerical formula] 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 [numerical formula] are obtained. Connection between dilogarithm identities and algebraic K-theory (torsion in K_3(R)) is discussed. Relations between crystal basis, branching functions b_λ^<kΛo>(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^<kΛo>(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.
- 理論物理学刊行会の論文
- 1995-06-26
著者
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Kirillov Anatol
Department Of Mathematical Sciences University Of Tokyo : Steklov Mathematical Institute
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Kirillov Anatol
Department of Mathematical Sciences, University of Tokyo : Steklov Mathematical Institute