Discrete integrable systems, positivity, and continued fraction rearrangements

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.

24 pages, 2 figures

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• Noncommutative integrability, paths and quasi-determinants
• The solution of the $A_r$ T-system for arbitrary boundary
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Cited by in corpus (10)

• Discrete integrable systems and Poisson algebras from cluster maps
• Cluster algebras and continued fractions
• Webs on surfaces, rings of invariants, and clusters
• Motzkin numbers out of Random Domino Automaton
• Hankel determinant solutions to several discrete integrable systems and the Laurent Property
• The Coxeter relations and KP map for non-commuting symbols
• Non-commutative double-sided continued fractions
• A family of linearisable recurrences with the Laurent property
• Linear relations for Laurent polynomials and lattice equations
• Discrete Dynamical Systems From Real Valued Mutation