Magic squares: symmetry and combinatorics

Abstract

Doubly stochastic, magic square, and alternating sign matrices are matrices of order n over the set of real positive numbers , the set of nonnegative numbers , and the set of integers {−1,0,1}, respectively, having fixed row and columns sums of 1, an arbitrary positive integer N, and 1. Each can be expressed as a sum over permutation matrices of order n with coefficients that belong to , to the positive integers , and to , respectively. Mathematically, these objects are basic in combinatorics; physically, they arise in several contexts that are briefly reviewed. Little has been developed on their expansions in terms of permutation matrices, and little is known about counting formulas for them, except for alternating sign matrices where a closed formula for arbitrary n was recently obtained through the work of Zeilberger. Expansions of these matrices in terms of permutation matrices can be used to investigate and develop their properties. Such an expansion is called a representation of the magic square. Representations are, however, not unique, and the problem arises of enumerating the number of distinct representations of one and the same magic square. The present investigation addresses this problem in the context of primitive magic squares, which are defined as the class of magic squares of order n having a unique representation in which each permutation matrix occurs exactly once in the expansion, and such that this uniqueness is destroyed by the addition of another permutation matrix not already in the representation, a property called completeness. The set of primitive magic squares has a rich structure that is invariant under the action of a group G that is isomorphic to the dihedral group. The group G is definitive in unveiling the general structure of primitive magic squares by providing a complete labelling scheme that utilizes the (n − 4)-fold direct product group of G and a binary tree that specifies a path that shows how the elements in the direct product group are to be selected. Based on this structure, a recurrence relation is derived that generates all inequivalent primitive magic squares. The recurrence relation itself shows a hidden structure of more basic magic squares, called universal kernels, that underlie the structural form of all primitive magic squares. The recurrence relation for primitive magic squares is thus shifted to a recurrence relation for the universal kernels that is simpler in form, which is also derived. These recurrence relations produce one and the same primitive magic square in multiple ways, and the sorting out of the distinct magic squares thus generated remains a problem that is not yet solved.