[Geom., Alg. & Phys.] Eventual Sign Coherence for Quivers
ABSTRACT
The theory of cluster algebras has shown that many important spaces in math and physics have beautiful fundamental properties, such as the Laurent phenomenon, positivity, and sign coherence. This last property says that the combinatorial operation of mutation preserves some local structure in certain quivers, and the only known proofs rely on heavy tools from representation theory or algebraic geometry. Gekhtman and Nakanishi posed the Asymptotic Sign Coherence Conjecture for arbitrary quivers, which says sign coherence should eventually emerge in any sufficiently generic infinite mutation sequence. We prove, using purely combinatorial methods, that this conjecture holds with probability 1 for a random mutation sequence and that it holds in full for many families of quivers. This is joint work with Scott Neville. (https://sites.google.com/view/gapkias)
