The Tropical Totally Positive Grassmannian

Abstract

Tropical algebraic geometry is the geometry of the tropical semiring (?, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop+ Grk,n. We show that Trop+ Gr2,n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type An?3) associahedron, and that Trop+ Gr3,6 and Trop+ Gr3,7 are closely related to the fans dual to the types D4 and E6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types An?3, D4, and E6 for Gr2,n, Gr3,6, and Gr3,7. We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.