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A cluster expansion formula (\(A_{n}\) case). (English) Zbl 1184.13064
Cluster algebras were introduced by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] in order to study the dual canonical basis of a quantized enveloping algebra and the phenomenon of total positivity. Cluster algebras of finite type are classified by the Dynkin diagrams. The article under review gives a formula for the expansion of an arbitrary cluster variable in a cluster algebra of type A (or Ptolemy cluster algebra) in terms of a fixed initial seed. The formula is in terms of paths in the triangulation (of a regular polygon) corresponding to the seed.

13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
16S99 Associative rings and algebras arising under various constructions
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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