Commun. Korean Math. Soc. Published online December 29, 2021
Bokhee Im and Jonathan D. H. Smith
Chonnam National University; Iowa State University
Abstract : Fix an integer $n\ge 1$. Then the simplex $\Pi_n$, Birkhoff polytope $\Omega_n$, and Latin square polytope $\Lambda_n$ each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of $\Omega_n$ are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.
