Tropical $F$-polynomials and General Presentations

TL;DR

This paper introduces tropical $F$-polynomials for quiver representations, explores their geometric and combinatorial properties, and applies these concepts to cluster algebras and representation theory.

Contribution

It develops the theory of tropical $F$-polynomials, provides algorithms for Newton polytopes, and connects these to duality pairings and cluster structures.

Findings

Introduces tropical $F$-polynomials for quiver representations.

Provides an algorithm for determining Newton polytopes.

Establishes connections with duality pairings and cluster algebra structures.

Abstract

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra. We give an interpretation of evaluating $f_M$ at a weight vector. As a consequence, we give a presentation of the Newton polytope ${\sf N}(M)$ of $M$. We study the dual fan and 1-skeleton of ${\sf N}(M)$. We propose an algorithm to determine the generic Newton polytopes, and show 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.