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Towards a uniform subword complex description of acyclic finite type cluster algebras. (English) Zbl 1423.13118
Summary: 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.
MSC:
13F60 Cluster algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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