Maximal subgroups of the Coxeter group W(H4) and quaternions

Mehmet Koca; Ramazan Koç; Muataz Al Barwani; Shadia Al-Farsi

doi:10.1016/j.laa.2005.07.018

Maximal subgroups of the Coxeter group W(H₄) and quaternions

Mehmet Koca, Ramazan Koç, Muataz Al Barwani, Shadia Al-Farsi

Research output: Contribution to journal › Article › peer-review

Abstract

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H₄) of order 14,400. Its derived subgroup is the largest finite subgroup W(H₄)/Z₂ 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(H₂) × W(H₂)]⋊ Z₄ and W(H₃) × Z₂ possess non-crystallographic 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) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H₄) with sets of quaternion pairs acting on the quaternionic root systems.

Original language	English (US)
Pages (from-to)	441-452
Number of pages	12
Journal	Linear Algebra and Its Applications
Volume	412
Issue number	2-3
DOIs	https://doi.org/10.1016/j.laa.2005.07.018
State	Published - Jan 15 2006
Externally published	Yes

Keywords

Coxeter groups
Quaternions
Structure of groups
Subgroup structure

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

Access to Document

10.1016/j.laa.2005.07.018

Cite this

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title = "Maximal subgroups of the Coxeter group W(H4) and quaternions",

abstract = "The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 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(H2) × W(H2)]⋊ Z4 and W(H3) × Z2 possess non-crystallographic 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) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems.",

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T1 - Maximal subgroups of the Coxeter group W(H4) and quaternions

AU - Koca, Mehmet

AU - Koç, Ramazan

AU - Al Barwani, Muataz

AU - Al-Farsi, Shadia

PY - 2006/1/15

Y1 - 2006/1/15

N2 - The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 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(H2) × W(H2)]⋊ Z4 and W(H3) × Z2 possess non-crystallographic 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) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems.

AB - The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 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(H2) × W(H2)]⋊ Z4 and W(H3) × Z2 possess non-crystallographic 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) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems.

KW - Coxeter groups

KW - Quaternions

KW - Structure of groups

KW - Subgroup structure

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