Dihedral Transportation and (0,1)-Matrix Classes

TL;DR

This paper explores the existence of matrices with specific symmetry properties under dihedral group actions, extending classical transportation and (0,1)-matrix problems with new algebraic and combinatorial conditions.

Contribution

It introduces conditions for the existence of symmetric matrices invariant under dihedral subgroups, generalizing classical transportation and (0,1)-matrix results.

Findings

Conditions for invariant matrices under dihedral subgroup actions

Extension of Gale-Ryser Theorem to symmetric matrix classes

Constructive proofs for existence of such matrices

Abstract

Let R and S be two vectors of real numbers whose entries have the same sum. In the transportation problems one wishes to find a matrix A with row sum vector R and column sum vector S. If, in addition, the two vectors only contain nonnegative integers then one wants the same to be true for A. This can always be done and the transportation algorithm gives a method for explicitly calculating A. We can restrict things even further and insist that A have only entries zero and one. In this case, the Gale-Ryser Theorem gives necessary and sufficient conditions for A to exist and this result can be proved constructively. One can let the dihedral group D_4 of the square act on matrices. Then a subgroup of D_4 defines a set of matrices invariant under the subgroup. So one can consider analogues of the transportation and (0,1) problems for these sets of matrices. For every subgroup, we give conditions equivalent to the existence of the desired type of matrix.