Towards a uniform subword complex description of acyclic finite type cluster algebras

Brodsky, Sarah B.¹ ; Stump, Christian¹

¹ Institut für Mathematik, Technische Universität Berlin, Germany

Algebraic Combinatorics, Tome 1 (2018) no. 4, pp. 545-572.

Résumé

It has been established in recent years how to approach acyclic cluster algebras of finite type using subword complexes. We continue this study by uniformly describing the $c$- and $g$-vectors, and by providing a conjectured description of the Newton polytopes of the $F$-polynomials. We moreover show that this conjectured description would imply that finite type cluster complexes are realized by the duals of the Minkowski sums of the Newton polytopes of either the $F$-polynomials or of the cluster variables, respectively. We prove this conjectured description to hold in type $A$ and in all types of rank at most $8$ including all exceptional types, leaving types $B$, $C$, and $D$ conjectural.

Détail

Reçu le : 2017-12-20
Accepté le : 2018-06-12
Publié le : 2018-09-10
DOI : 10.5802/alco.25

Keywords: cluster algebra, $F$-polynomial, subword complexes

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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