Alternating subgroups of Coxeter groups and their spinor extensions

TL;DR

This paper provides presentations for the alternating subgroup and its spinor cover of any Coxeter group, and applies the Coxeter--Todd algorithm to specific types, advancing understanding of their algebraic structures.

Contribution

It introduces a general presentation for the groups $G^+$ and $ ilde{G}^+$ based on Coxeter graphs, and applies the Coxeter--Todd algorithm to types A, B, and D.

Findings

Presentations for $G^+$ and $ ilde{G}^+$ for any Coxeter system.

Application of Coxeter--Todd algorithm to types A, B, D.

Enhanced understanding of the algebraic structure of Coxeter groups.

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.