The structure of the full icosahedral group

We consider the full icosahedral group, which is the Coxeter group of type H_3. The Coxeter groups appear naturally in geometry and algebra. In 1935, the finite Coxeter groups were classified by Coxeter in terms of Coxeter-Dynkin diagrams. We remark that the affine extensions of the Coxeter groups of types H are related to quasicrystals with tenfold symmetry. Our approach to understanding the structure of Coxeter groups is the noncommutative Groebner basis theory, which is called the Groebner-Shirshov basis theory. By completing the relations coming from a presentation of the Coxeter group, we find a Groebner-Shirshov basis to obtain a set of standard monomials. Especially, for the Coxeter group of type H_3, its Groebner-Shirshov basis and the corresponding standard monomials are constructed. Thus, we understand the algebra structure of the group algebra C[H_3], which is not commutative.