Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E_8 Root System

Vertices of the 4-dimensional semi-regular polytope, $\mathit{\text{snub}}\mathit{\text{24-cell}}$ and its symmetry group $W(D_4):C_3$ of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of $E_8$ root system. The quaternionic root system of $H_4$ splits as the vertices of 24-cell and the \textit{snub 24-cell} under the symmetry group of the $\mathit{\text{snub}}\mathit{\text{24-cell}}$ which is one of the maximal subgroups of the group $W(H_4)$ as well as $W(F_4)$. It is noted that the group is isomorphic to the$$semi-direct product of the Weyl group of $D_4$ with the cyclic group of order 3 denoted by $W(D_4):C_3$, the Coxeter notation for which is $[3,4,3^{+}]$. We analyze the vertex structure of the $\mathit{\text{snub}}\mathit{\text{24-cell}}$ and decompose the orbits of $W(H_4)$ under the orbits of $W(D_4):C_3$. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group $W(D_4):C_3$. In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group $W(D_4):C_3$. The dual polytope of the $\mathit{\text{snub}}\mathit{\text{24-cell}}$ is explicitly constructed. Decompositions of the Archimedean $W(H_4)$ polytopes under the symmetry of the group $W(D_4):C_3$ are given in the appendix.