F-Polynomials of Donaldson-Thomas Transformations

Abstract

$F$-polynomials are integer coefficient polynomials encoding the mutations of cluster variables inside a cluster algebra. In this article, we study the $F$-polynomials associated with the action of Donaldson-Thomas transformations on cluster variables. For acyclic quivers, quivers of surface types, and quivers associated with triples of flags, we give explicit descriptions of their Donaldson-Thomas $F$-polynomials in terms of generating functions for ideals inside a labeled poset. We also describe the combinatorial procedure needed to modify these labeled posets to obtain Donaldson-Thomas $F$-polynomials for full subquivers and triangular extensions.

27 pages, 19 figures