On systems of orthogonal permutations
The authors introduce the notion of a complete system of orthogonal permutations in the \(n\)-dimensional real vector space \({\mathbb R}^{n}\): A linear transformation \(P:{\mathbb R}^{n}\to {\mathbb R}^{n}\) is called a (linear) permutation if for any \({\vec a}=(a_{1},\ldots,a_{n})\in{\mathbb R}^{n}, \;P{\vec a}=(\varepsilon_{1}{(P)}a_{\pi_{P}(1)}, \ldots , \varepsilon_{n}{(P)}a_{\pi _{P}(n)}),\) where \(\pi_{P}\in S_{n}\) is a permutation and \(\varepsilon_{i}(P)\) is \(+1\) or \(-1\); two permutations \(P\) and \(P'\) are called orthogonal if the scalar product \((P\vec a, P'\vec a)=0\) for every \(\vec{a}\in {\mathbb R}^{n};\) a set \(P_{1},\ldots,P_{n}\) of permutations is said to be complete if \(P_{i}\) and \(P_{j}\) are orthogonal for \(i,j=1,\ldots,n\), \(i\not=j.\) To each complete system of orthogonal permutations \(P_{1},\ldots, P_{n}\) one associates an \(n\)-dimensional division algebra over \(\mathbb R\) (not necessarily associative) and from the well-known result about real finite dimensional division algebras it follows that there exists no complete system of orthogonal permutations if \(n\not= 1,2,4,8.\) This gives an answer to a problem announced by B. Sendov. In case of dimensions \(n=2,4,8\) the structure of orthogonal permutations is obtained which corresponds to the complex number field, the skew field of quaternions and the Cayley algebra.
scientific article; zbMATH DE number 1523752