Discrete Power Functions on a Hexagonal Lattice I: Derivation of defining equations from the symmetry of the Garnier System in two variables

TL;DR

This paper derives the defining equations of a discrete power function on a hexagonal lattice from the symmetry properties of the Garnier system in two variables, linking discrete geometry with integrable systems.

Contribution

It establishes a novel connection between the symmetry of the Garnier system and the defining equations of the discrete power function on a hexagonal lattice.

Findings

Derived the defining equations from Garnier system symmetry

Linked discrete power functions to integrable systems

Provided a new geometric interpretation of the equations

Abstract

The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discrete symmetry of the Garnier system in two variables.