Automorphisms of graph-universal Coxeter groups
This paper deals with the group \(\Aut(W)\) of automorphisms of a graph-universal type Coxeter group. Recall that a Coxeter diagram \(M=(m_{i,j})_{i,j\in I}\) over a set \(I\) is called graph-universal if \(m_{i,j}\in\{2,\infty\}\) for all \(i,j\in I\). Consider a Coxeter group \(W\) associated with such a diagram. Then \(W\) acts on the conjugacy classes of involutions of \(W\). Let \(\text{Spe}(W)\) denote the kernel of this action, and let \(\text{Inn}(W)\) denote the group of inner automorphisms of \(W\). It was shown by \textit{J. Tits} [J. Algebra 113, No. 2, 346-357 (1988; Zbl 0657.20027)]\ that \(\Aut(W)\) is a semidirect extension of \(\text{Spe}(W)\) with some finite group. In this paper, the author gives an explicit presentation of \(\text{Spe}(W)\), and deduces a necessary and sufficient condition for \(\text{Spe}(W)\) to be equal to \(\text{Inn}(W)\).
scientific article; zbMATH DE number 1139793