Positivity through Broken Lines

1. I N T R O D U C T I O N (1.1) Cluster algebras are a class of commutative algebras defined by a three step process. The first step is to specify an initial seed s, that is some set of initial generators of an ambient field F . Second, a subset X of F is constructed by mutating the initial seed s, mutating the resulting seeds, mutating the result of those mutations, and so forth ad infinitum. Finally, the cluster algebra here is the subalgebra of the ambient field F that is generated by the set X . This construction is perhaps rather intimidating. I certainly was alarmed when I first saw this definition. I wondered how anything could possibly be proven for such objects! But perhaps the main point is that many cluster algebras that appear in nature come with a geometric or combinatorial model in which seeds are certain distinguished bases or coordinates or something of this sort, the process of mutation is the natural way to change from one basis of coordinate system to another. Indeed, Fomin and Zelevinsky, the founders of the theory, were precisely trying to understand ideas around canonical bases and total positivity. For a detailed history, see the introduction to their seminal paper [FZ02]. (1.2) Perhaps the first indication that cluster algebras are possibly very well-behaved comes from the Laurent Phenomenon. To explain this, first an interlude on mutation. Beginning with a set of generators {x0 i } m 1 ··= {x0 1 , . . . , x0 m} of a field F , mutation in direction k produces a new set of generators {x1 i } m 1 . For i 6= k, no change occurs: x1 i = x0 i . However, x1 k and x0 k are related via the exchange relation (1.2.1) x k x 0 k = M1 +M2 for certain coprime monomials M1 and M2 in the other variables. Clearly, x1 k is a Laurent polynomial in the original variables {x0 i } m 1 with positive coefficients. Now imagine mutating the new set of variables in some direction ` 6= k to get variables {x2 i }0 and then mutating again in direction k to finally obtain a set of variables {x3 i } m 1 . Pictorially: {x1 1 , . . . , x 1 m} {x 3 1 , . . . , x 3 m}. {x0 1 , . . . , x 0 m} {x2 1 , . . . , x 2 m} ` k k Date: December 14, 2016. 1