9 research outputs found
Six types of functions of the Lie groups O(5) and G(2)
New families of -functions are described in the context of the compact
simple Lie groups O(5) and G(2). These functions of two real variables
generalize the common exponential functions and for each group, only one family
is currently found in the literature. All the families are fully characterized,
their most important properties are described, namely their continuous and
discrete orthogonalities and decompositions of their products.Comment: 25 pages, 13 figure
The rings of n-dimensional polytopes
Points of an orbit of a finite Coxeter group G, generated by n reflections
starting from a single seed point, are considered as vertices of a polytope
(G-polytope) centered at the origin of a real n-dimensional Euclidean space. A
general efficient method is recalled for the geometric description of G-
polytopes, their faces of all dimensions and their adjacencies. Products and
symmetrized powers of G-polytopes are introduced and their decomposition into
the sums of G-polytopes is described. Several invariants of G-polytopes are
found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers
and congruence classes of the polytopes. The definitions apply to
crystallographic and non-crystallographic Coxeter groups. Examples and
applications are shown.Comment: 24 page