Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 012, 64 pages      arXiv:1805.04197      https://doi.org/10.3842/SIGMA.2019.012

The Kashaev Equation and Related Recurrences

Alexander Leaf
Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

Received May 24, 2018, in final form February 03, 2019; Published online February 21, 2019

Abstract
The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.

Key words: Kashaev equation; hexahedron recurrence; principal minors of symmetric matrices; cubical complexes; s-holomorphicity; cluster algebras.

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