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{\bf Philippe Di Francesco and Rinat Kedem}
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{\bf Positivity of the T-System Cluster Algebra}
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We give the path model solution for the cluster algebra variables of
the $T$-system of type $A_r$ with generic boundary conditions. The
solutions are partition functions of (strongly) non-intersecting paths
on weighted graphs. The graphs are the same as those constructed for
the $Q$-system in our earlier work, and depend 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 which
distinguishes the $T$-system from the $Q$-system, usually identified
as the spectral parameter in the context of representation theory. The
path model is alternatively described on a graph with non-commutative
weights, and cluster mutations are interpreted as non-commutative
continued fraction rearrangements. As a consequence, the solution is a
positive Laurent polynomial of the seed data.

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