Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS

  • Im, Bokhee (Department of Mathematics Chonnam National University) ;
  • Smith, Jonathan D.H. (Department of Mathematics Iowa State University)
  • Received : 2021.04.15
  • Accepted : 2021.09.02
  • Published : 2022.04.30
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Abstract

Fix an integer n ≥ 1. Then the simplex Πn, Birkhoff polytope Ωn, and Latin square polytope Λn 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 Ωn 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

Acknowledgement

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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