Extending Upper Cluster Algebras

Abstract

Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose that there is some cluster variable $x_e$ such that $R_{x_e}=S[x_e^{\pm 1}]$. We try to understand under which conditions $R$ is an upper cluster algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the restriction of $(\Delta ,W)$ to some subquiver is a cluster model, we give a sufficient condition for $(\Delta ,W)$ itself being a cluster model. As an application, we show that the semi-invariant ring of any complete $m$-tuple flags is an upper cluster algebra whose quiver is explicitly given. Moreover, the quiver with its rigid potential is a polyhedral cluster model.

27 pages,7 figures. Comments are welcome

References in corpus (1)

• Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$