|XXVIII Workshop on Geometric Methods in Physics||28.06-04.07.2009|
Discretization of tori of exceptional compact simple Lie groups
We consider an exceptional simple Lie group G and a positive integer M and introduce a certain finite set of lattice points F_M. The group G and corresponding affine Weyl group induce the symmetry of F_M, the number M determines the density of the grid F_M. We present a construction of the set F_M and count the numbers of its points |F_M|. The relation between the numbers |F_M| and numbers of elements of finite order in G is discussed. We specify the maximal sets of pairwise orthogonal C- and S-functions. These finite sets allow us to calculate Fourier like discrete expansions of arbitrary discrete functions on F_M. Application of these discrete transforms to interpolation is presented on the cases G_2, F_4 and E_8.