Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$

Details Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$ Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$

2006-06-30

arxiv.org/abs/math/0606775v1

Mathematics

Rings and Algebras

4 pages, no figures

Scientific paper

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 obtained by P.Caldero and the author in math.RT/0604054. 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 in math.CO/0602408. The arguments in math.RT/0604054 used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick, self-contained and completely elementary alternative proof of the same results.

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