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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.

MSC:
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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