Latin squares and Hurwitz theorem
A classical theorem of Hurwitz states that only for \(n\in\{1,2,4,8\}\) is it possible to write \[ \left(\sum_{i=1}^n x_i^2\right)\left(\sum_{i=1}^n y_i^2\right) =\sum_{i=1}^n z_i^2 \] where each \(z_i\) is a (complex) linear combination of terms of the form \(x_ay_b\) where \(1\leq a,b\leq n\). The multiplicativity of the norms on the complex numbers, quaternions and octonions provide the required identities for \(n=2,4,8\) respectively. Hurwitz's argument generalises to any field of characteristic other than \(2\). The present paper provides a proof of the special case of Hurwitz's theorem where each \(z_i\) is an \textit{integer} linear combination of terms of the form \(x_ay_b\). The interesting feature of the proof is that it is combinatorial in nature. In essence, it reduces the Hurwitz problem to an existence question for a particular kind of signed Latin square. A Latin square is a matrix in which each row and column is a permutation of the same set of symbols. The author seems not to be familiar with the literature on Latin squares; they invent new names and reprove some things that are well known in the field. For example, the ``symmetric'' Latin squares defined before Example 1 are more usually called ``totally symmetric'' Latin squares. More crucially, in established terminology, an ``intercalate'' is a \(2\times2\) submatrix which is itself a Latin square. A Latin square of order \(n\) cannot have more than \(\frac{1}{4}n^2(n-1)\) intercalates, which is how many it will have if each pair of cells that share a common symbol are in an intercalate. The author calls Latin squares with this property ``well-painted''. He spends several pages proving that the only well-painted Latin squares are isotopic to (i.e. can be obtained by permuting rows, columns and/or symbols of) the Cayley table of the elementary abelian \(2\)-group (but without recognising it as such). This result was published explicitly in this form by \textit{K. Heinrich} and \textit{W. D. Wallis} [Lect. Notes Math. 884, 221--233 (1981; Zbl 0475.05014)] and various equivalent statements had been proved earlier. See [\textit{J. M. Browning} et al., J. Comb. Theory, Ser. A 124, 41--56 (2014; Zbl 1283.05040)] some details and a generalisation to larger subsquares. After that, the paper gets more interesting and original. Extra information is imposed by giving each entry in a Latin square a sign (either \(+\) or \(-\)). A signed Latin square is then said to be \textit{polarized} if (its unsigned version) is well-painted and each of the intercalates gets given an odd number of minus signs. It is proved that polarized Latin squares only exist for orders \(n\in\{1,2,4,8\}\). This follows from the non-existence of an example of order 16, given that the polarized property will clearly be inherited by Latin subsquares. Finally, it is shown that the Hurwitz problem only has solutions over the integers if there exists a polarized Latin square of the appropriate order.
scientific article; zbMATH DE number 6473707