
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 doubledimer 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 sholomorphicity, 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; sholomorphicity; cluster algebras.
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References

Chelkak D., Smirnov S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math. 189 (2012), 515580, arXiv:0910.2045.

Felsner S., Geometric graphs and arrangements. Some chapters from combinatorial geometry, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Wiesbaden, 2004.

Fomin S., Pylyavskyy P., Shustin E., Morsifications and mutations, arXiv:1711.10598.

Kashaev R.M., On discrete threedimensional equations associated with the local YangBaxter relation, Lett. Math. Phys. 38 (1996), 389397, arXiv:solvint/9512005.

Kenyon R., Pemantle R., Principal minors and rhombus tilings, J. Phys. A: Math. Theor. 47 (2014), 474010, 17 pages, arXiv:1404.1354.

Kenyon R., Pemantle R., Doubledimers, the Ising model and the hexahedron recurrence, J. Combin. Theory Ser. A 137 (2016), 2763, arXiv:1308.2998.

Kozlov D., Combinatorial algebraic topology, Algorithms and Computation in Mathematics, Vol. 21, Springer, Berlin, 2008.

Leaf A., The Kashaev equation and related recurrences, Ph.D. Thesis, University of Michigan, 2018.

Manin Yu.I., Schechtman V.V., Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, in Algebraic Number Theory, Adv. Stud. Pure Math., Vol. 17, Academic Press, Boston, MA, 1989, 289308.

Oeding L., Settheoretic defining equations of the variety of principal minors of symmetric matrices, Algebra Number Theory 5 (2011), 75109, arXiv:0809.4236.

Wilson R.J., Introduction to graph theory, Longman, Harlow, 1996.

Ziegler G.M., Higher Bruhat orders and cyclic hyperplane arrangements, Topology 32 (1993), 259279.

