The Chevalley group G_{2}(2) of order 12096 and the octonionic root system of E_{7}

The octonionic root system of the exceptional Lie algebra E_8 has been constructed from the quaternionic roots of F_4 using the Cayley-Dickson doubling procedure where the roots of E_7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E_7 is the adjoint Chevalley group G_2(2) of order 12096. One of the four maximal subgroups of G_2(2) of order 192 preserves the quaternion subalgebra of the E_7 root system. The other three maximal subgroups of orders 432,192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E_6xU(1), SU(2)xSO(12), and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension

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)

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 emphasizedComment: 26 pages, 10 figure

Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A_4 projects onto only two prototiles: thick and thin rhombuses of the Penrose tilings. Similarly the Delone cells tiling the same root lattice projects onto two isosceles Robinson triangles which also lead to Penrose tilings with kites and darts. We point out that the Coxeter element of order h=n+1 and the dihedral subgroup of order 2n plays a crucial role for h-fold symmetric aperiodic tilings of the Coxeter plane. After setting the general scheme we give examples leading to tilings with 4-fold, 5-fold, 6-fold,7-fold, 8-fold and 12-fold symmetries with rhombic and triangular tilings of the plane which are useful in modelling the quasicrystallography with 5-fold, 8-fold and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A_(3)whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4 fold symmetry.Comment: 22 pages, 17 figure