On Superalgebras of Matrices with Symmetry Properties

TL;DR

This paper explores how various symmetry properties in square matrices lead to superalgebra structures, unifying different symmetry types and providing explicit formulas for their construction.

Contribution

It introduces a unifying framework linking multiple matrix symmetries to superalgebra structures and derives explicit representations for these matrices.

Findings

Semi-magic square matrices form a superalgebra with symmetry-based grading.

Other symmetries also induce superalgebra structures, revealing new symmetry types.

Explicit formulas enable construction of matrices with these symmetries.

Abstract

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complementary parts. In particular, this provides a unifying framework for the composite `most perfect square' symmetry and the related class of `reversible squares'; moreover, the semi-magic square algebra is identified as part of a 2-gradation of the general square matrix algebra. We derive explicit representation formulae for matrices of all symmetry types considered, which can be used to construct all such matrices.