Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements

Research output: Contribution to journalReview articlepeer-review


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 Ar. 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.

Original languageEnglish (US)
Pages (from-to)299-324
Number of pages26
JournalLetters in Mathematical Physics
Issue number1-3
StatePublished - Jun 2011
Externally publishedYes


  • Laurent phenomenon
  • cluster algebras
  • continued fractions
  • integrable systems
  • non-commutative
  • positivity

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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