Discrete integrable systems, positivity, and continued fraction rearrangements

Details Discrete integrable systems, positivity, and continued fraction rearrangements Discrete integrable systems, positivity, and continued fraction rearrangements

2010-09-09

arxiv.org/abs/1009.1911v1

Physics

Mathematical Physics

24 pages, 2 figures

Scientific paper

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.

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