Quaterionic Construction of the W(F_4) Polytopes with Their Dual Polytopes and Branching under the Subgroups B(B_4) and W(B_3)*W(A_1)

TL;DR

This paper constructs 4D F4 polytopes and their duals using quaternionic representations of Coxeter groups, analyzing their structure and subgroup branchings with a focus on group theory and quaternion techniques.

Contribution

It introduces a quaternionic approach to construct and analyze F4 polytopes, their duals, and subgroup branchings, providing new insights into their geometric and algebraic properties.

Findings

Quaternionic construction of F4 polytopes and duals.

Branching patterns under B4 and B3×A1 subgroups.

Emphasis on group theoretical techniques and quaternion use.

Abstract

4-dimensional $F_{4} $ polytopes and their dual polytopes have been constructed as the orbits of the Coxeter-Weyl group $W(F_{4})$ where the group elements and the vertices of the polytopes are represented by quaternions. Branchings of an arbitrary \textbf{$W(F_{4})$} orbit under the Coxeter groups $W(B_{4} $ and $W(B_{3}) \times W(A_{1})$ have been presented. The role of group theoretical technique and the use of quaternions have been emphasized