Orthogonal bases for transportation polytopes applied to Latin squares, magic squares and Sudoku boards

TL;DR

This paper introduces a straightforward method to construct orthogonal bases for matrix spaces related to transportation polytopes, Latin squares, magic squares, and Sudoku boards, utilizing tensor products and binary trees.

Contribution

It provides a novel, simple construction of orthogonal bases for these combinatorial and geometric structures, respecting various symmetry properties.

Findings

Constructed orthogonal bases for matrices with zero row and column sums.

Bases are compatible with symmetries like centrosymmetry and skew-centrosymmetry.

Method applies to Birkhoff polytope, contingency tables, Latin squares, magic squares, and Sudoku boards.

Abstract

We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric pieces; the bases can be easily modified to respect the usual matrix symmetry and skew-symmetry as well.