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 finite band of lattice lines, and we ask when the projection fills every lattice point of that band. In affine type, boundedness is equivalent to ${\delta }_v={\delta }_w$ for the null root $\delta$. Writing this common coordinate as $r$, we prove that a band line fills exactly when its transjective root classes cover every residue modulo $r$. This yields a complete classification for every acyclic orientation of every simply-laced affine diagram. For a source-sink ${\tilde{A}}_n$ quiver, every projection fills except the source-sink diagonal. In tree type, every width-one band fills, while every width-two band fails on its two boundary lines because of a congruence gap. The Auslander--Reiten defect further determines whether such a boundary line is finite or contains an infinite arithmetic progression; finiteness is governed by a balanced-geodesic criterion. On the annulus, the absolute defect is the crossing number with the core curve, and the $\delta$-shift is the Dehn twist along it.

23 pages, 1 figure; ancillary exact-arithmetic verification scripts included