2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134746We evaluate the Geiss-Leclerc-Schroer $ϕ$-map for shape modules over the preprojective algebra $Λ$ of type $\hat{A_1}$ in terms of matrix minors arising from the block-Toeplitz representation of the loop group $\SL_2(\mathcal{L})$. Conjecturally these minors are among the cluster variables for coordinate rings of unipotent cells within $\SL_2(\mathcal{L})$. In so doing we compute the Euler characteristic of any generalized flag variety attached to a shape module by counting standard tableaux of requisite shape and parity; alternatively by counting chess tableaux of requisite shape and content.Representation TheoryCombinatorics22E67; 05E10; 15A57; 16G20; 14L40; 05E05text