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We evaluate the Geiss-Leclerc-Schroer $\phi$-map for shape modules over thepreprojective algebra $\Lambda$ of type $\hat{A_1}$ in terms of matrix minorsarising from the block-Toeplitz representation of the loop group$\SL_2(\mathcal{L})$. Conjecturally these minors are among the clustervariables for coordinate rings of unipotent cells within $\SL_2(\mathcal{L})$.In so doing we compute the Euler characteristic of any generalized flag varietyattached to a shape module by counting standard tableaux of requisite shape andparity; alternatively by counting chess tableaux of requisite shape andcontent.

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