Lens partition function, pentagon identity, and star-triangle relation

Star - Triangle Relation for a Three Dimensional Model

The solvable sl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising type model on the body centered cubic lattice with twoand threespin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation,...

A PARTITION IDENTITY AND THE UNIVERSAL MOCK THETA FUNCTION g2

We prove analytic and combinatorial identities reminiscent of Schur’s classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts satisfy congruence conditions. Furthermore, if small parts are excluded, the resulting overpartitions are generated by the product of a modular form and Gor...

Pentagon Relation for the Quantum Dilogarithm and Quantized M

Steinhaus’ Problem on Partition of a Triangle

H. Steinhaus has asked whether inside each acute triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. We present two solutions of Steinhaus’ problem. The n-dimensional case of Theorem 1 below was proved in [6], see also [2] and [4, Theorem 2.1, p. 152]. For an earlier mass-partition version of Theorem 1, for bounded convex masses ...

Graphical Approach to the Nonintersecting String Model: Star-triangle Equation, Inversion Relation, and Exact Solution

A general q-state nonintersecting string model for q Z 2 is studied, and solved, using a graphical approach. This model, of which special cases were first introduced by Stroganov and Schultz, has recently drawn further attention because of its relation to Potts models. In its simplest version, a uniform, separable nonintersecting string model on a quadratic lattice, it corresponds to a q*-state...