TY - JOUR

T1 - Non-commutative integrability, paths and quasi-determinants

AU - Di Francesco, Philippe

AU - Kedem, Rinat

N1 - Funding Information:
We would like to thank S. Fomin for discussions, and L. Faddeev for pointing out Refs. [12, 13] to us. We thank the Mathematisches ForschungsInstitut Oberwolfach, Germany, Research in Pairs program (Aug. 2009) during which this work was initiated. P.D.F. also thanks S. Fomin for hospitality at the Dept. of Mathematics of the University of Michigan, Ann Arbor (Spring 2010). R.K. thanks the Institut Henri Poincaré, Paris, France, for hospitality during the semester “Statistical Physics, Combinatorics and Probability” (Fall 2009), and the Institut de Physique Théorique du CEA Saclay, France. P.D.F. received partial support from the ANR Grant GranMa, the ENIGMA research training network MRTN-CT-2004-5652, and the ESF program MISGAM. R.K. is supported by NSF grant DMS-0802511.

PY - 2011/9/10

Y1 - 2011/9/10

N2 - In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.

AB - In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.

KW - Cluster algebra

KW - Continued fractions

KW - Discrete integrable systems

KW - Noncommutative

KW - Path models

KW - Quantum cluster algebra

KW - Quasi-determinants

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U2 - 10.1016/j.aim.2011.05.017

DO - 10.1016/j.aim.2011.05.017

M3 - Article

AN - SCOPUS:79960296922

VL - 228

SP - 97

EP - 152

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 1

ER -