Discrete integrable systems, positivity, and continued fraction rearrangements

TL;DR

This paper reviews a unified approach to solving discrete integrable systems, demonstrating that solutions can be expressed as positive Laurent polynomials through continued fraction rearrangements, applicable even in non-commutative cases.

Contribution

It introduces a novel reformulation of mutations as continued fraction rearrangements that preserve positivity, extending to non-commutative systems.

Findings

Solutions are Laurent polynomials with non-negative coefficients.

Reformulation preserves positivity in both commutative and non-commutative settings.

Techniques unify various discrete integrable systems.

Abstract

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.