2026/06/30 by Sarah B. Brodsky
Mathematics · #math.CO #math.RA #math.RT #msc:13F60 #msc:05E10 #msc:16G20
23 pages, 1 figure; ancillary exact-arithmetic verification scripts included
arxiv created 2026/07/30 · arxiv updated 2026/07/31
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 (cv, cw) of this configuration: when the difference cv - cw is bounded, the image lies in a finite band of lattice lines, and we ask when the projection fills that band. In finite type, boundedness is automatic; in affine type we prove that it is equivalent to equality of the corresponding null-root coordinates, δv = δw. We then resolve the filling problem for affine type \widetildeAn in the source-sink orientation: every coordinate projection fills its band except for the source-sink pair, whose diagonal contains only the finite regular part. More generally, we classify the non-filling banded pairs for every acyclic orientation of every affine diagram: apart from the source-sink diagonals in type \widetildeA, non-filling occurs only at boundary lines reached by a unique extremal real root, and precisely when the geodesic joining the two vertices is balanced. The obstruction is the Auslander--Reiten defect; on the annulus this picture is topological: the defect is the crossing number with the core curve, and the δ-shift is the Dehn twist along it.