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Abstract: We give the path model solution for the cluster algebra variables of the$A r$ $T$-system with generic boundary conditions. The solutions are partitionfunctions of strongly non-intersecting paths on weighted graphs. The graphsare the same as those constructed for the $Q$-system in our earlier work, anddepend on the seed or initial data in terms of which the solutions are given.The weights are -time-dependent- where -time- is the extra parameter whichdistinguishes the $T$-system from the $Q$-system, usually identified as thespectral parameter in the context of representation theory. The path model isalternatively described on a graph with non-commutative weights, and clustermutations are interpreted as non-commutative continued fraction rearrangements.As a consequence, the solution is a positive Laurent polynomial of the seeddata.



Author: P. Di Francesco, R. Kedem

Source: https://arxiv.org/







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