Generalising Latin square orthogonality and Frobenius-König with alternating sign matrices

Abstract

The theory of Latin squares has a long history. While the objects themselves appeared earlier, the study of their general mathematical theory dates back to Euler in the 18th century. Latin squares can be interpreted as 3-dimensional permutation hypermatrices, and alternating sign matrices often arise as a natural generalisation of permutation matrices. In 2018, Brualdi and Dahl introduced a generalisation of classical Latin squares using alternating sign hypermatrices. Inspired by their definition, we develop the theory of Italian squares, a related generalisation of Latin squares, together with a notion of orthogonality that resolves an inconsistency in the definition of Brualdi and Dahl. Building on classical questions from Latin square theory, we obtain results including upper bounds on the maximal size of a pairwise orthogonal set, conditions for the existence of an orthogonal mate, infinite families of orthogonal pairs, and transversals. As part of our exploration of alternating sign matrices, we also prove a Frobenius-König type result for a class of $(0,\pm1)$-matrices.