Generic variables in acyclic cluster algebras |
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Authors: | G Dupont |
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Institution: | Université de Lyon, Université Lyon 1, Institut Camille Jordan - UMR 5208 du CNRS, 43, Bd du 11 novembre 1918, 69622 Villeurbanne cedex, France |
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Abstract: | Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q). We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig’s dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set G(Q) is a Z-basis of A(Q) and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases. |
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Keywords: | 13F60 16G20 |
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