Block-Toeplitz determinants, chess tableaux, and the type $\hat{A_1}$ Geiss-Leclerc-Schroer $φ$-map

We evaluate the Geiss-Leclerc-Schroer $\phi$-map for shape modules over the preprojective algebra $\Lambda$ 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.