Solutions to the T-systems with Principal Coefficients

TL;DR

This paper introduces solutions to T-systems with principal coefficients using combinatorial methods, connecting cluster algebra, perfect matchings, paths, and networks, and extends to various related systems.

Contribution

It defines T-systems with principal coefficients from a cluster algebra perspective and provides combinatorial solutions applicable to multiple related systems.

Findings

Provides combinatorial solutions for T-systems with principal coefficients

Connects T-systems solutions to perfect matchings, paths, and networks

Extends solutions to systems like octahedron recurrence and pentagram maps

Abstract

The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer's octahedron recurrence (Speyer 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2014).