Combinatorics of $F$-polynomials

Abstract

We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the $F$-polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of the $F$-polynomial is saturated for any rigid representation. We provide many examples and counterexamples, and pose several conjectures.

25 pages; V2. modification according to arXiv:1911.10513; V3. various corrections and modifications thanks to the referee's great effort

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