Geodesic
Semicanonical basis generators of the cluster algebra of type \(A_1^{(1)}\).
The electronic journal of combinatorics, Tome 14 (2007)
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We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp. The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton. This note provides a quick, self-contained and completely elementary alternative proof of the same results.
DOI : 10.37236/1005
Classification : 16G20, 05E99
Mots-clés : cluster variables, semicanonical bases, cluster algebras, closed formula, Laurent expansions, Laurent polynomials 
@article{10_37236_1005,
     author = {Andrei Zelevinsky},
     title = {Semicanonical basis generators of the cluster algebra of type {\(A_1^{(1)}\).}},
     journal = {The electronic journal of combinatorics},
     year = {2007},
     volume = {14},
     doi = {10.37236/1005},
     zbl = {1144.16015},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1005/}
}
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Andrei Zelevinsky. Semicanonical basis generators of the cluster algebra of type \(A_1^{(1)}\).. The electronic journal of combinatorics, Tome 14 (2007). doi: 10.37236/1005

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