Mathematics – Representation Theory
Scientific paper
2009-10-28
Mathematics
Representation Theory
26 pages. Section 4 was slightly changed according to the comments of the referee. The rest of the paper remains essentially u
Scientific paper
Sherman-Zelevinsky and Cerulli constructed canonically positive bases in cluster algebras associated to affine quivers having at most three vertices. Both constructions involve cluster monomials and normalized Chebyshev polynomials of the first kind evaluated at a certain "imaginary" element in the cluster algebra. Using this combinatorial description, it is possible to define for any affine quiver $Q$ a set $\mathcal B(Q)$ which is conjectured to be the canonically positive basis of the acyclic cluster algebra $\mathcal A(Q)$. In this article, we provide a geometric realization of the elements in $\mathcal B(Q)$ in terms of the representation theory of $Q$. This is done by introducing an analogue of the Caldero-Chapoton cluster character where the usual quiver Grassmannian is replaced by a constructible subset called transverse quiver Grassmannian.
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