Maximal Subgroups of the Coxeter Group $W(H_4)$ and Quaternions

TL;DR

This paper classifies the maximal subgroups of the noncrystallographic Coxeter group W(H4), detailing their structures, orders, and quaternionic representations, thereby advancing understanding of symmetries in higher-dimensional spaces.

Contribution

It provides a comprehensive classification of the maximal subgroups of W(H4) and introduces quaternionic representations for these subgroups.

Findings

Identified five non-normal maximal subgroups of W(H4)

Connected subgroup structures to Weyl groups of classical Lie groups

Represented subgroups using quaternion pairs acting on root systems

Abstract

The largest finite subgroup of O(4) is the noncrystallographic Coxeter group $W(H_{4})$ of order 14400. Its derived subgroup is the largest finite subgroup $W(H_{4})/Z_{2}$ of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups $[ W(H_{2})\times W(H_{2})] \times Z_{4}$ and $W(H_{3})\times Z_{2}$ possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of $SU(3)\times SU(3)$%, SU(5) and SO(8) respectively. We represent the maximal subgroups of $% W(H_{4})$ with sets of quaternion pairs acting on the quaternionic root systems.