Pairwise orthogonal Italian squares: transversal criteria, and new entries at orders 6 and 7

Abstract

An Italian square of order n is the compressed presentation of an alternating sign hypermatrix, and two of them are orthogonal when every plane of one meets every plane of the other in Frobenius inner product 1. Ernst, Lia, O'Brien, Sheekey and Zumbrägel introduced this notion of orthogonality, proved that a pairwise orthogonal set (a POIS) of order n has at most n-1 members, and closed with a table recording, for each order n ≤ 7 and each i, the largest POIS with exactly i non-Latin members; several of its entries are lower bounds or blanks. Applying the partial-sum description of an alternating sign matrix along the third axis, we obtain that an alternating sign hypermatrix is exactly a chain 0 = Q₀, Q₁, …, Qₙ = J of (0,1)-matrices whose successive differences are alternating sign matrices, that orthogonality telescopes along that chain, and that the first plane of such a hypermatrix is a permutation matrix. The last of these turns the common-transversal argument the authors use for one order-4 pair into a criterion that tests a pair, and then into one that tests a single square. Using them we raise the (7,1) entry of the table from ≥ 2 to ≥ 3 by exhibiting three pairwise orthogonal Italian squares of order 7 exactly one of which is not Latin; and we decide two instances of its open (6,3) cell, showing that the order-6 pair the authors print is maximal — no Italian square of order 6, Latin or not, is orthogonal to both members — and exhibiting a non-Latin Italian square of order 6 that lies in no set of three POIS. Every theorem stated here is machine-checked in Lean 4.

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