Alternating subgroups of Coxeter groups and their spinor extensions

Abstract

Let $G$ be a discrete Coxeter group, $G^{+}$ its alternating subgroup and ${\tilde{G}}^{+}$ the spinor cover of $G^{+}$. A presentation of the groups $G^{+}$ and ${\tilde{G}}^{+}$ is proved for an arbitrary Coxeter system $(G,S)$; the generators are related to edges of the Coxeter graph. Results of the Coxeter--Todd algorithm - with this presentation - for the chains of alternating groups of types A, B and D are given.