Communications of the
Korean Mathematical Society CKMS

Fix an integer $nge 1$. 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.

Keywords: Birkhoff polytope, stochastic matrix, quasigroup, Latin square, approximate quasigroup, generalized doubly stochastic, tristochastic tensor

MSC numbers: 15B51, 05B15, 20N05, 51F20, 52B12

Supported by: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (NRF-2017R1D1A3B05029924).