Nakajima's quiver varieties and triangular bases of rank-2 cluster algebras

TL;DR

This paper proves the support conjecture for all skew-symmetric rank-2 cluster algebras, confirming that the support of triangular basis elements is bounded by a specific region, advancing understanding of their structure.

Contribution

It establishes the support conjecture for skew-symmetric rank-2 cluster algebras, a significant step in the theory of quantum cluster algebras.

Findings

Support of triangular basis elements is bounded by an explicitly described region.

The support conjecture is proven for all skew-symmetric rank-2 cluster algebras.

Advances understanding of the structure of quantum cluster algebras.

Abstract

Berenstein and Zelevinsky introduced quantum cluster algebras [Adv. Math, 2005] and the triangular bases [IMRN, 2014]. The support conjecture by Lee-Li-Rupel-Zelevinsky [PNAS, 2014] asserts that the support of a triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region that is possibly concave. In this paper, we prove the support conjecture for all skew-symmetric rank-2 cluster algebras.