Tropical F$F$‐polynomials and general presentations

Abstract

AbstractWe introduce the tropical ‐polynomial of a quiver representation . We study its interplay with the general presentation for any finite‐dimensional basic algebra. We give an interpretation of evaluating at a weight vector. As a consequence, we give a presentation of the Newton polytope of . We study the dual fan and 1‐skeleton of . We propose an algorithm to determine the generic Newton polytopes, and show that it works for path algebras. As an application, we give a representation‐theoretic interpretation of Fock–Goncharov's duality pairing. We give an explicit construction of dual clusters, which consists of real Schur representations. We specialize the above general results to the cluster‐finite algebras and the preprojective algebras of Dynkin type.