A cluster expansion formula ($A_n$ case)

Details A cluster expansion formula ($A_n$ case) A cluster expansion formula ($A_n$ case)

2006-11-30

arxiv.org/abs/math/0611956v2

Mathematics

Representation Theory

9 pages, 2 figures; Version 2: condition (T6) added in Definition 1, Section 2 rewritten, references added

Scientific paper

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\zS$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\zS$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\zS$.

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