Monomial bases for the primitive complex Shephard groups of rank three

Abstract

For the group algebra of each primitive non-Coxeter Shephard group of rank three, we construct a monomial basis and its explicit multiplication table. First, we find a Gröbner–Shirshov basis for the Shephard group of type L 2. Then, since each of the groups of types L 3 and M 3 has a parabolic subgroup isomorphic to the group of type L 2, by using the sets of minimal right coset representatives of L 2 in L 3 and M 3, respectively, we apply the Gröbner–Shirshov basis technique to find the monomial bases for the Shephard groups of rank three L 3 and M 3. From this, we obtain the operation tables between the elements in each of the groups. Also we explicitly show that the group of type M 3 has a subgroup of index 2 isomorphic to the group of type L 3.