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 2008, Di Francesco and Kedem 2009). 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 2007), generalized lambda-determinants (Di Francesco 2013) and (higher) pentagram maps (Schwartz 1992, Ovsienko et al. 2010, Glick 2011, Gekhtman et al. 2016).
Highlights
The A∞ T-system [DFK13], called the octahedron recurrence, is a discrete dynamical system of formal variables Ti,j,k for i, j, k ∈ Z satisfyingTi,j,k−1Ti,j,k+1 = Ti−1,j,kTi+1,j,k + Ti,j−1,kTi,j+1,k.This recurrence relation preserves the parity of i + j + k and there are two independent systems depending on the parity of i + j + k
We show that the cluster variables satisfy the recurrence relation (4) on {Ti,j,k | (i, j, k) ∈ Z3odd} with an extra set of coefficients {ci,j | (i, j) ∈ Z2}
We show that our network and elementary network matrices coincide with the objects studied in [DFK13] in the case of coefficient-free T-systems (ci,j = 1 for all (i, j) ∈ Z2)
Summary
The A∞ T-system [DFK13], called the octahedron recurrence, is a discrete dynamical system of formal variables Ti,j,k for i, j, k ∈ Z satisfying. This gives another form of the perfect-matching solution (Theorem 5.4): Ti0,j0,k0 =. This bijection can be extended to G.
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