Geodesic
A cluster expansion formula (\(A_{n}\) case)
The electronic journal of combinatorics, Tome 15 (2008)
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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 $\Sigma$ 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 $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.
DOI : 10.37236/788
Classification : 13F60, 16G20, 16S99
Mots-clés : cluster algebra, cluster variable, regular polygon, Ptolemy cluster algebra, mutation, triangulation 
@article{10_37236_788,
     author = {Ralf Schiffler},
     title = {A cluster expansion formula {(\(A_{n}\)} case)},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/788},
     zbl = {1184.13064},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/788/}
}
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Ralf Schiffler. A cluster expansion formula (\(A_{n}\) case). The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/788

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