2003/12/15 by David Speyer, Speyer, David, Lauren K. Williams +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO
paper · pdf · doi:10.48550/arxiv.math/0312297
22 pages, 10 figures
arxiv created 2003/12/15 · arxiv updated 2009/12/01
Tropical algebraic geometry is the geometry of the tropical semiring (R, 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 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.