TY - JOUR
T1 - Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements
AU - Di Francesco, Philippe
N1 - Funding Information:
We would like to thank especially R. Kedem for a fruitful collaboration. We also thank S. Fomin for hospitality at the Dept. of Mathematics of the University of Michigan, Ann Arbor, while this work was completed. We received partial support from the ANR Grant GranMa, the ENIGMA research training network MRTN-CT-2004-5652, and the ESF program MISGAM.
PY - 2011/6
Y1 - 2011/6
N2 - 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.
AB - 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.
KW - Laurent phenomenon
KW - cluster algebras
KW - continued fractions
KW - integrable systems
KW - non-commutative
KW - positivity
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U2 - 10.1007/s11005-010-0429-x
DO - 10.1007/s11005-010-0429-x
M3 - Review article
AN - SCOPUS:79954910873
SN - 0377-9017
VL - 96
SP - 299
EP - 324
JO - Letters in Mathematical Physics
JF - Letters in Mathematical Physics
IS - 1-3
ER -