zbMATH — the first resource for mathematics

Non-commutative integrability, paths and quasi-determinants. (English) Zbl 1252.37069
Summary: 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 [A. Berenstein and A. Zelevinsky, Adv. Math. 195, No. 2, 405–455 (2005; Zbl 1124.20028)], the Kontsevich evolution [P. Di Francesco and R. Kedem, Int. Math. Res. Not. 2010, No. 21, 4042–4063 (2010; Zbl 1276.16025)] and the \(T\)-systems themselves [P. Di Francesco and R. Kedem, Electron. J. Comb. 16, No. 1, Research Paper R140, 39 p. (2009: Zbl 1229.13019)]. 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.

37K60 Lattice dynamics; integrable lattice equations
13F60 Cluster algebras
Full Text: DOI arXiv
[1] Assem, I.; Reutenauer, C.; Smith, D., Frises, preprint
[2] Bazhanov, V.; Reshetikhin, N., Restricted solid-on-solid models connected with simply laced algebras and conformal field theory, J. phys. A, 23, 1477-1492, (1990) · Zbl 0712.17024
[3] Berenstein, A.; Zelevinsky, A., Quantum cluster algebras, Adv. math., 195, 405-455, (2005) · Zbl 1124.20028
[4] Caldero, P.; Zelevinsky, A., Laurent expansions in cluster algebras via quiver representations, Mosc. math. J., 6, 3, 411-429, (2006) · Zbl 1133.16012
[5] Di Francesco, P., The solution of the \(A_r\)T-system with arbitrary boundary, (2010), preprint
[6] Di Francesco, P.; Kedem, R., Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property, Lett. math. phys., 89, 3, 183-216, (2009) · Zbl 1195.81077
[7] Di Francesco, P.; Kedem, R., Q-systems, heaps, paths and cluster positivity, Comm. math. phys., 293, 3, 727-802, (2009) · Zbl 1194.05165
[8] Di Francesco, P.; Kedem, R., Positivity of the T-system cluster algebra, Electron. J. combin., 16, 1, R140, (2009), Oberwolfach preprint OWP 2009-21 · Zbl 1229.13019
[9] Di Francesco, P.; Kedem, R., Q-system cluster algebras, paths and total positivity, SIGMA symmetry integrability geom. methods appl., 6, 014, (2010), 36 pp. · Zbl 1241.13020
[10] Di Francesco, P.; Kedem, R., Discrete non-commutative integrability: proof of a conjecture by M. Kontsevich, Int. math. res. not. IMRN, (2010) · Zbl 1276.16025
[11] Etingof, P.; Retakh, V., Quantum determinants and quasideterminants, Asian J. math., 3, 345-351, (1999) · Zbl 0985.17013
[12] Faddeev, L.D.; Kashaev, R.M.; Volkov, A.Y., Strongly coupled quantum discrete Liouville theory, I. algebraic approach and duality, Comm. math. phys., 219, 199-219, (2001) · Zbl 0981.81052
[13] Faddeev, L.D.; Volkov, A.Y., Discrete evolution for the zero modes of the quantum Liouville model, J. phys. A, 41, 19, 194008, (2008), 12 pp. · Zbl 1139.81385
[14] Fomin, S.; Green, C., Noncommutative Schur functions and their applications, Discrete math., 306, 10-11, 1080-1096, (2006) · Zbl 1096.05051
[15] Fomin, S.; Zelevinsky, A., Cluster algebras, I, J. amer. math. soc., 15, 2, 497-529, (2002) · Zbl 1021.16017
[16] Fomin, S.; Zelevinsky, A., The Laurent phenomenon, Adv. in appl. math., 28, 2, 119-144, (2002) · Zbl 1012.05012
[17] Frenkel, E.; Reshetikhin, N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, (), 163-205 · Zbl 0973.17015
[18] Gelfand, I.; Gelfand, S.; Retakh, V.; Wilson, R.L., Quasideterminants, Adv. math., 193, 1, 56-141, (2005) · Zbl 1079.15007
[19] Gelfand, I.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 2, 218-348, (1995) · Zbl 0831.05063
[20] Gessel, I.M.; Viennot, X., Binomial determinants, paths and hook formulae, Adv. math., 58, 300-321, (1985) · Zbl 0579.05004
[21] Kedem, R., Q-systems as cluster algebras, J. phys. A, 41, 194011, (2008), 14 pp. · Zbl 1141.81014
[22] Kirillov, A.N.; Reshetikhin, N.Yu., Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. sov. math., 52, 3156-3164, (1990)
[23] M. Kontsevich, private communication.
[24] Kuniba, A.; Nakanishi, T.; Suzuki, J., Functional relations in solvable lattice models, I. functional relations and representation theory, Internat. J. modern phys. A, 9, 5215-5266, (1994) · Zbl 0985.82501
[25] Lindström, B., On the vector representations of induced matroids, Bull. lond. math. soc., 5, 85-90, (1973) · Zbl 0262.05018
[26] Musiker, G.; Propp, J., Combinatorial interpretations for rank-two cluster algebras of affine type, Electron. J. combin., 14, (2007), R15 · Zbl 1140.05053
[27] Nakajima, H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. theory, 7, 259-274, (2003) · Zbl 1078.17008
[28] Sherman, P.; Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. math. J., 4, 4, 947-974, (2004) · Zbl 1103.16018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.