Department of

August 2015 September 2015 October 2015 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 4 5 1 2 3 2 3 4 5 6 7 8 6 7 8 9 10 11 12 4 5 6 7 8 9 10 9 10 11 12 13 14 15 13 14 15 16 17 18 19 11 12 13 14 15 16 17 16 17 18 19 20 21 22 20 21 22 23 24 25 26 18 19 20 21 22 23 24 23 24 25 26 27 28 29 27 28 29 30 25 26 27 28 29 30 31 30 31

Wednesday, September 9, 2015

**Abstract:** The $A_\infty$ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient free cluster algebra [Kedem 08]. We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perfect matchings, non-intersecting paths and networks. This also provides a solution to other systems with various choices of coefficients on T-systems including Speyer’s octahedron recurrence [Speyer 07], generalized lambda-determinants [Di Francesco 13] and (higher) pentagram maps [Schwartz 92, Ovsienko et al. 00, Glick 11, Gekhtman et al. 14].