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In this paper the root polytopes of all finite reflection groups W with a connected Coxeter–Dynkin diagram in 
are identified, their faces of dimensions 0 ≤ d ≤ n − 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter–Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566–584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.
are identified, their faces of dimensions 0 ≤ d ≤ n − 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter–Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566–584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.
