Tropical $F$-polynomials and Cluster Algebras

Abstract: The representation-theoretic interpretations of $g$-vectors and $F$-polynomials are two fundamental ingredients in the (additive) categorification of cluster algebras. We knew that the $g$-vectors are related to the presentation spaces. We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$, and explain its interplay with the general presentation for any finite-dimensional basic algebra. As a consequence, we give a presentation of the Newton polytope $N(M)$ of $M$. 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 cluster duality pairing. We also study many combinatorial aspects of $N(M)$, such as faces, the dual fan and $1$-skeleton. We conjecture that the coefficients of a cluster monomial corresponding to vertices are all $1$, and the coefficients inside the Newton polytope are saturated. We show the conjecture holds for acyclic cluster algebras. We specialize the above general results to the cluster-finite algebras and the preprojective algebras of Dynkin type.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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