Length functions and Demazure operators for G(e,1,n), I

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function l(W) associated to a given generator set S is completely described by the function n(W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space HW in terms of Demazure operators, and show that HW is isomorphic to the coinvariant algebra SW, which enables us to define a homogeneous basis on SW parametrized by w ϵ W.