Geometric permutations
The paper is concerned with so called geometric permutations. It starts by showing the connection between permutations and Coxeter-Bennet configurations (CBC's are regular graphs on \(d+3\) vertices having degree \(d\)). This connection is used to define an equivalence relation on permutations. Next some properties of difference permutations (roughly \(i_j - i_{j+1}\) where \(i_0 i_1 \dots i_{d+2}\) is the permutation) are given. It is shown that CBC-equivalence can be determined from the difference permutations. Next a permutation is defined to be geometric if the CBC-configuration is a so called orthoscheme. Then a step number \(k\) corresponding to a permutation is introduced. The main theorem of the paper states that a permutation is geometric if and only if the step number \(k\) is either 0, 1 or \(d+1\). For the definition of orthoscheme the author refers to earlier publications. The following details of the paper caused some concern: The step number is defined for normed permutations but is used much broader, which requires some explanation. In Section 3 below formula (7) it is stated that for type 2: \(0<DV<1\). I fail to see how this follows from \(0<s<m<t\) or \(s>m>t>0\) (\(DV = DV(0,s,t,m) = \frac{t}{t-s}\frac{m}{m-s}\)). On page 342, 12th line from below the \(\pi_1\) should be a \(\pi_l\).
scientific article; zbMATH DE number 5911451