\(Q\)-systems, heaps, paths and cluster positivity. (English) Zbl 1194.05165
The \(A_r\) \(Q\)-system is a recursion relation satisfied by characters of special irreducible finite-dimensional modules of the Lie algebra \(A_r\). It is a discrete integrable dynamical system. The relations of the \(Q\)-system are mutations in a cluster algebra. The authors prove the positivity property of the corresponding cluster variables, by using the integrability property. The system can be mapped to several different types of statistical models: path models, heaps on graphs, or domino tilings. The choice of the initial conditions for the recursion relations determines the specific model. The authors construct an explicit solution of the \(Q\)-system as a function of any possible set of initial conditions.
Reviewer: Florin Nicolae (Berlin)
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 13F60 | Cluster algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |
| 37A60 | Dynamical aspects of statistical mechanics |
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