Invited by School of Mathematics and Statistics, Prof. Jiarui Fei from Shanghai Jiao Tong University will give a lecture online.

Title: Tropical $F$-polynomials and Cluster Algebras

Time: 15:00 p.m.,Mar. 29th, 2021

Conference ID: 844 381 815( Tencent Conference) , PW:125914

Web Site: https://meeting.tencent.com/s/2cHfLPZzvB81

Abstract: The $g$-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 $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra.

To be more specific, we give an interpretation of evaluating $f_M$ at a weight vector. 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 duality pairing.

We also study many combinatorial aspects of $N(M)$, such as 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.