Solutions to the T-systems with Principal Coefficients
Abstract
The 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).
Cite
@article{arxiv.1511.05535,
title = {Solutions to the T-systems with Principal Coefficients},
author = {Panupong Vichitkunakorn},
journal= {arXiv preprint arXiv:1511.05535},
year = {2023}
}