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

Brodsky, Sarah B.; Stump, Christian

doi:10.5802/alco.25

Brodsky, Sarah B.¹; Stump, Christian ¹

¹ Institut für Mathematik, Technische Universität Berlin, Germany

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

Abstract

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.

Received: 2017-12-20
Accepted: 2018-06-12
Published online: 2018-09-10

MR Zbl

DOI: 10.5802/alco.25

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

Author's affiliations:

Brodsky, Sarah B. ¹; Stump, Christian ¹

¹ Institut für Mathematik, Technische Universität Berlin, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Towards a uniform subword complex description of acyclic finite type cluster algebras},
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Brodsky, Sarah B.; Stump, Christian. Towards a uniform subword complex description of acyclic finite type cluster algebras. Algebraic Combinatorics, Volume 1 (2018) no. 4, pp. 545-572. doi : 10.5802/alco.25. https://alco.centre-mersenne.org/articles/10.5802/alco.25/

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