Geodesic
Positivity of the \(T\)-system cluster algebra
The electronic journal of combinatorics, Tome 16 (2009) no. 1

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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.
DOI : 10.37236/229
Classification : 13F60, 17B37, 05C90, 17B80 
Philippe Di Francesco; Rinat Kedem. Positivity of the \(T\)-system cluster algebra. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/229
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     author = {Philippe Di Francesco and Rinat Kedem},
     title = {Positivity of the {\(T\)-system} cluster algebra},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {1},
     doi = {10.37236/229},
     zbl = {1229.13019},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/229/}
}
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