Latin squares and Hurwitz theorem
type: Item
instance of: scholarly article
title: Latin squares and Hurwitz theorem
zbMATH Open document ID: 1348.11032
author: Serge Yagunov
published in: Expositiones Mathematicae
publication date: 2015-08-19T00:00:00Z
review text: 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. Walli…
reviewed by: Ian M. Wanless
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 6473707
zbMATH Keywords: Hurwitz theorem
zbMATH Keywords: Latin squares
zbMATH Keywords: intercalates
zbMATH Keywords: normed algebras
zbMATH Keywords: sums of squares
MaRDI profile type: Publication
full work available at URL: j.exmath.2014.12.002
OpenAlex ID: W2080002020
cites work: The octonions
cites work: Q3993060
DOI: 10.1016/J.EXMATH.2014.12.002
description: scientific article; zbMATH DE number 6473707