Coordinate projections of $c$-vectors of cluster algebras from the annulus

Abstract

For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real Schur roots of the associated root system. We study the two-coordinate projections $(c_v, c_w)$ of this configuration: when the difference $c_v - c_w$ is bounded the image lies in a band of lattice lines, and we ask when the projection fills that band. A band-existence dichotomy, valid in every acyclic type, shows the difference is bounded if and only if the null root satisfies $δ_v = δ_w$. For affine type $\widetilde{A}_n$ (the annulus), in the source-sink orientation, we resolve the filling question completely: every coordinate projection fills its band except along the source-sink diagonal, which carries only the finite regular part. The obstruction is the Auslander--Reiten defect, which a projection sees on its diagonal exactly when the defect is a coordinate difference; the only such pair is the source-sink pair of $\widetilde{A}_n$, so the pattern depends on the chosen seed. More generally, every banded pair of null-root coefficient one fills, except these diagonals. Off the diagonal a banded pair in $\widetilde{E}_7$ fails to fill, so non-filling is not confined to type $\widetilde{A}_n$; a computation classifies the pairs of coefficient at least two over a range of affine types, where this $\widetilde{E}_7$ pair is the only further failure, and the general classification remains open.