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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)
##### Keywords:
cluster algebra; $$F$$-polynomial; subword complexes
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##### References:
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