The Simplistic Expression And Coset Decomposition Of The Elements In The Complex Reflection Group G(m, M, R)

In 1954,Shephard and Todd classified all finite irreducible complex reflection groups [1].After that,people pay more and more attention to Complex reflection groups.Complex reflection groups are generalizations of real reflection groups and there are many similar problems in the Complex reflection groups that have not been solved.For example,the reduced expressions of the elements in most Complex reflection groups are unknown.We will make an exploratory research on reduced expressions of the elements in imprimitive Complex reflection group G(m,m,r)in this paper.We give a set of complete right coset representatives of the subgroup G(m,m,r-1)in G(m,m,r),based on which we can obtain the right coset decompositions of the elements in group G(m,m,r);We give the quasi-reduced expressions of the elements in two special groups G(3,3,3)and G(4,4,3);We compare the right coset decompositions with quasi-reduced expressions of the elements in two groups G(3,3,3)and G(4,4,3).