Filters

20092

More filters

20192
Reset filters

Settings

Author: Di Francesco, Philippe × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × WGL Institute: MFO ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Discrete non-commutative integrability: The proof of a conjecture by M. Kontsevich
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Di Francesco, Philippe; Kedem, Rinat
    We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the ommutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case.
  • Thumbnail Image
    Item
    Positivity of the T-system cluster algebra
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Di Francesco, Philippe; Kedem, Rinat
    We give the path model solution for the cluster algebra variables of the Ar T-system 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 “timedependent” 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 utations are interpreted as non-commutative continued fraction rearrangements. As a consequence, the solution is a positive Laurent polynomial of the seed data.