Homogeneous conditions for stochastic tensors
Commun. Korean Math. Soc. 2022 Vol. 37, No. 2, 371-384
https://doi.org/10.4134/CKMS.c210127
Published online March 29, 2022
Printed April 30, 2022
Bokhee Im, 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.
Keywords : Birkhoff polytope, stochastic matrix, quasigroup, Latin square, approximate quasigroup, generalized doubly stochastic, tristochastic tensor
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).
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