On the minimal elements in conjugacy classes of the complex reflection group G(r,1,n)

Abstract

In this paper we study the minimal length elements in conjugacy classes (abbreviated as minimal elements) of the complex reflection group G(r,1,n). We set up some connections between the set of minimal elements in different conjugacy classes of G(r,1,n), which yields an recursive way to produce all minimal elements. For each w∈G(r,1,n), we give a necessary and sufficient condition under which w is minimal in terms of its standard components in the standard decomposition of w. We construct an explicit example which shows that the standard basis elements labeled by two different minimal elements in the same conjugacy class may have different images in the cocenter of the non-degenerate cyclotomic Hecke algebra of type G(r,1,n), which disproves an earlier speculation. Finally, we show that the standard basis elements labeled by some minimal elements form an integral basis for the cocenter of the degenerate cyclotomic Hecke algebra of type G(r,1,n).