Nakajima's quiver varieties and triangular bases of bipartite cluster algebras

TL;DR

This paper extends the understanding of triangular bases in bipartite cluster algebras by establishing support bounds, building on prior work that focused on rank-2 cases.

Contribution

It generalizes the support conjecture for triangular bases from rank-2 to bipartite cluster algebras, providing new bounds on their support regions.

Findings

Support bounds for triangular basis elements in bipartite cluster algebras are established.

The results extend previous support conjecture proofs from rank-2 to bipartite cases.

The work connects Nakajima's quiver varieties with cluster algebra bases.

Abstract

Berenstein and Zelevinsky introduced quantum cluster algebras \cite{BZ1} and the triangular bases \cite{BZ2}. The support conjecture proposed in \cite{LLRZ}, which asserts that the support of each triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region, was established in \cite{L} for skew-symmetric rank-2 cluster algebras. In this paper we extend this result by proving a bound on the support of each triangular basis element for bipartite cluster algebras.