Reflection Ordering on the Group

In the paper [11], we introduced a partial ordering, called the reflection ordering, on the elements of G(m, p, n) and described such an ordering on the groups G(m, 1, n), m > 1. In the present paper, we describe the reflection ordering on the group G(m, m, n). As a by-product, we obtain a formula for the enumeration of a certain subset in the symmetric group Sn, which is of independent combinatorial interest. §0. Introduction. Let P (respectively, N, Z) denote the set of positive integers (respectively, nonnegative integers, integers). For any k, n ∈ N with k 6 n, let [k, n] = {k, k + 1, ..., n} and [n] = [1, n]. 0.1. The group G(m,p,n). Fix m, p, n ∈ P with p | m, let G(m, p, n) be the group consisting of all n × n monomial matrices w such that all the non-zero entries, say θ1, ..., θn, of w are mth roots of unity with ( ∏n i=1 θi) m/p = 1. Any w ∈ G(m, p, n) can be expressed in the form w = [a1, ..., an | σ] with σ ∈ Sn and ak ∈ Z, where Sn is the symmetric group on the set [n], and the entry in the (k, (k)σ)-position of w is exp ( 2πak √−1 m ) for k ∈ [n]. Clearly, p |nk=1 ak . 0.2. Reflection groups. A reflection in a complex vector space V is by definition a linear transformation of V of finite order, whose fixed point subspace has codimension 1 in V . A reflection defined here is called a pseudo-reflection in some literature to distinguish it from the concept of a reflection in a euclidean space. A group