Abstract
We study Newton polytopes of cluster variables in type $A_n$ cluster algebras, whose cluster and coefficient variables are indexed by the diagonals and boundary segments of a polygon. Our main results include an explicit description of the affine hull and facets of the Newton polytope of the Laurent expansion of any cluster variable, with respect to any cluster. In particular, we show that every Laurent monomial in a Laurent expansion of a type $A$ cluster variable corresponds to a vertex of the Newton polytope. We also describe the face lattice of each Newton polytope via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. Nous étudions polytopes de Newton des variables amassées dans les algèbres amassées de type A, dont les variables sont indexés par les diagonales et les côtés d’un polygone. Nos principaux résultats comprennent une description explicite de l’enveloppe affine et facettes du polytope de Newton du développement de Laurent de toutes variables amassées. En particulier, nous montrons que tout monôme Laurent dans un développement de Laurent de variable amassée de type A correspond à un sommet du polytope de Newton. Nous décrivons aussi le treillis des facesde chaque polytope de Newton via un isomorphisme avec le treillis des sous-graphes élémentaires du “snake graph” qui est associé.
Highlights
Cluster algebras, introduced by Fomin and Zelevinsky in the early 2000’s [10], are a class of commutative rings equipped with a distinguished set of generators that are grouped into sets of constant cardinality n
Diagonals correspond to cluster variables, triangulations correspond to clusters, boundary segments correspond to coefficient variables, and mutation corresponds to a local move called a flip of the triangulation, in which one diagonal is replaced with another one
We study the Newton polytope of the Laurent expansion of a type A cluster variable with respect to an arbitrary cluster
Summary
Cluster algebras, introduced by Fomin and Zelevinsky in the early 2000’s [10], are a class of commutative rings equipped with a distinguished set of generators (cluster variables) that are grouped into sets of constant cardinality n (the clusters). The study of Newton polytopes of Laurent expansions of cluster variables was initiated by Sherman and Zelevinsky in their study of rank 2 cluster algebras, in which it was shown that the Newton polygon of any cluster variable in a rank 2 cluster algebra of finite or affine type is a triangle [25]. Our first main result in this paper is Theorem 3.12, a description of the face lattice of the Newton polytope of a Laurent expansion of any cluster variable of type An via an isomorphism with the lattice of elementary subgraphs of the associated snake graph. For full proofs of all results in this paper, see [15]
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