• Corpus ID: 53073827

# A formula for $F$-Polynomials in terms of $C$-Vectors and Stabilization of $F$-Polynomials

@article{Gupta2018AFF,
title={A formula for \$F\$-Polynomials in terms of \$C\$-Vectors and Stabilization of \$F\$-Polynomials},
author={Meghal Gupta},
journal={arXiv: Combinatorics},
year={2018}
}
• Meghal Gupta
• Published 5 December 2018
• Mathematics
• arXiv: Combinatorics
Given a quiver associated to a cluster algebra and a sequence of vertices, iterative mutation leads to $F$-Polynomials which appear in numerous places in the cluster algebraic literature. The coefficients of the monomials in these $F$-Polynomials are difficult to understand and have been an area of study for many years. In this paper, we present a general closed-form formula for these coefficients in terms of elementary manipulations with $C$-matrices. We then demonstrate the effectiveness of…
5 Citations
Newton polytopes of rank 3 cluster variables
• Mathematics
• 2019
We characterize the cluster variables of skew-symmetrizable cluster algebras of rank 3 by their Newton polytopes. The Newton polytope of the cluster variable $z$ is the convex hull of the set of all
Geometric description of C-vectors and real L\"osungen
• Mathematics
• 2019
We propose a combinatorial/geometric model and formulate several conjectures to describe the c-matrices of an arbitrary skew-symmetrizable matrix. In particular, we introduce real L\"osungen as an
F-POLYNOMIALS FOR THE R-KRONECKER QUIVER
Definition 1.1 ((Framed) Quiver). A quiver is a directed graph with no 2-cycles, where multiple edges are allowed. Given a quiver Q with n vertices labelled {1, 2, . . . , n}, the corresponding
Earthquake theorem for cluster algebras of finite type
• Mathematics
• 2022
. We introduce a cluster algebraic generalization of Thurston’s earthquake map for the cluster algebras of ﬁnite type, which we call the cluster earthquake map . It is deﬁned by gluing exponential
Two Formulas for $F$-Polynomials
• Mathematics
• 2021
We discuss a product formula for F -polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for F polynomials. The other is based on the

## References

SHOWING 1-10 OF 22 REFERENCES
Canonical bases for cluster algebras
• Mathematics
• 2014
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical
On Cluster Variables of Rank Two Acyclic Cluster Algebras
In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix $${{\left(\begin{array}{ll} 0 &c \\ A combinatorial formula for rank 2 cluster variables • Mathematics • 2011 AbstractLet r be any positive integer, and let x1,x2 be indeterminates. We consider the sequence {xn} defined by the recursive relation$$x_{n+1} =\bigl(x_n^r +1\bigr)/{x_{n-1}} for any integer n.
Cluster algebras and derived categories
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review
Stable Cluster Variables
Richard Eager and Sebastian Franco introduced a change of basis transformation on the F-polynomials of Fomin and Zelevinsky, corresponding to rewriting them in the basis given by fractional brane
Cluster expansion formulas and perfect matchings
• Mathematics
• 2008
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these
On Cluster Algebras Arising from Unpunctured Surfaces
• Mathematics
• 2007
We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables
Donaldson-Thomas theory and cluster algebras
We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in
Quivers with potentials and their representations I: Mutations
• Mathematics
• 2007
Abstract.We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of