Minimal nonstandard Coxeter trees
A Coxeter graph \(\Gamma\) consists of a set of vertices \(S\) and an associated matrix \(M\). The Witt form of \(\Gamma\) is a symmetric bilinear form \(B_ \Gamma\) defined over the real vector space \(V_ \Gamma\) of dimension \(| \Gamma |\) by \(B_ \Gamma = -\cos (\pi/m_{s,t})\) if \(m_{s,t} \neq+\infty\) and \(=-1\) if \(m_{s,t} = + \infty\). The signature of \(\Gamma\) is the \(l\)-tuple \((\varepsilon_ 1, \dots, \varepsilon_ l)\) where \(\varepsilon_ i\) is \(-1,0\) or 1 such that there exists \(P \in \text{GL} (V_ \Gamma)\) with \(PB_ \Gamma P^ t=\text{diag} (\varepsilon_ 1, \dots, \varepsilon_ l)\). A Coexter graph is nonstandard if its signature does not have 1's in all but the first position. The paper studies the structure of minimal nonstandard Coxeter trees, where minimal means not containing another Coxeter tree as a minor. A Coxeter graph is called ordinary if the weight of each edge is 3. The paper includes a complete list of ordinary minimal nonstandard Coxeter trees.
scientific article; zbMATH DE number 495540