Let $ \Bbb{N} $ (respectively, $ \Bbb{Z} $, $ \Bbb{R} $, $ \Bbb{C} $) be the set of all positive integers (respectively, integers, real numbers, complex numbers). Let $ V $ be a Hermitian space of dimension $ n $. A reflection in $ V $ is a linear transformation of $ V $ of finite order with exactly $ n-1 $ eigenvalues equal to $ 1 $. A reflection group $ G $ on $ V $ is a finite group generated by reflections in $ V $. A reflection group $ G $ is called a Coxeter group if there is a $ G $-invariant $ \Bbb{R} $-subspace $ V_0 $ of $ V $ such that the canonical map $ \Bbb{C} \otimes_\Bbb{R} V_0 \rightarrow V $ is bijective, or $ G $ is called a complex group. A reflection group $ G $ on $ V $ is called imprimitive if $ V $ is a direct sum of nontrivial linear subspaces $ V = V_1 \oplus V_2 \oplus \cdots \oplus V_t $ such that every element $ w \in G $ is a permutation on the set $ \{V_1, V_2, \cdots, V_t\} $.

For any $ m, p, r \in \Bbb{N} $ with $ p \mid m $ (read "$ p $ divides $ m $"), let $ G(m, p, r) $ be the group consisting of all $ r\times r $ monomial matrices whose non-zero entries $ a_1, a_2, \cdots, a_r $ are $ m $th roots of unity with $ \left(\prod^r_{i = 1}a_i\right)^{m/p} = 1 $, where $ a_i $ is in the $ i $-th row of the monomial matrix. In [1], Shephard and Todd proved that any irreducible imprimitive reflection group is isomorphic to some $ G(m, p, r) $. We see that $ G(m, p, r) $ is a Coxeter group if either $ m\leq2 $ or $ (p, r) = (m, 2) $.

The imprimitive reflection group $ G(m, p, r) $ can also be defined by a presentation $ (S, P) $, where $ S $ is a set of generators of $ G(m, p, r) $, subject only to the relations in $ P $. In the cases $ p = 1 $, $ p = m $, and $ 1 < p < m $, we list their presentations as follows (see [2]).

(1) When $ p = 1 $, $ S $ contains $ r $ reflections $ s_0 $ and $ s_i $ for $ i \in \{1, 2, \cdots, r-1\} $, and $ P $ consists of the relations $ s_0^m = s_i^2 = 1 $ for $ i \in \{1, 2, \cdots, r-1\} $; $ s_is_{i+1}s_i = s_{i+1}s_is_{i+1} $ for $ i \in \{1, 2, \cdots, r-2\} $; $ s_is_j = s_js_i $ for $ i, j \in \{0, 1, 2, \cdots, r-1\} $ and $ |i-j|>1 $; $ s_0s_1s_0s_1 = s_1s_0s_1s_0 $.

(2) When $ p = m $, $ S $ contains $ r $ reflections $ s'_1 $ and $ s_i $ for $ i \in \{1, 2, \cdots, r-1\} $, and $ P $ consists of the relations $ {s'_1}^2 = s_i^2 = 1 $ for $ i \in \{1, 2, \cdots, r-1\} $; $ s_is_{i+1}s_i = s_{i+1}s_is_{i+1} $ for $ i \in \{1, 2, \cdots, r-2\} $; $ s_is_j = s_js_i $ for $ i, j \in \{1, 2, \cdots, r-1\} $ and $ |i-j|>1 $; $ s'_1s_i = s_is'_1 $ for $ i>2 $; $ s'_1s_2s'_1 = s_2s'_1s_2 $; $ \underbrace{s_{1}s'_{1}s_{1} \cdots}_{m} = \underbrace{s'_{1}s_{1}s'_{1}\cdots}_{m} $; $ s'_1s_1s_2s'_1s_1s_2 = s_2s'_1s_1s_2s'_1s_1 $.

(3) When $ 1 < p < m $, $ S $ contains $ r+1 $ reflections $ s_0, s'_1 $ and $ s_i $ for $ i \in \{1, 2, \cdots, r-1\} $, and $ P $ consists of the relations $ s_0^{m/p} = {s'_1}^2 = s_i^2 = 1 $ for $ i \in \{1, 2, \cdots, r-1\} $; $ s_is_{i+1}s_i = s_{i+1}s_is_{i+1} $ for $ i \in \{1, 2, \cdots, r-2\} $; $ s_is_j = s_js_i $ for $ i, j \in \{0, 1, 2, \cdots, r-1\} $ and $ |i-j|>1 $; $ s'_1s_i = s_is'_1 $ for $ i>2 $; $ s'_1s_2s'_1 = s_2s'_1s_2 $; $ \underbrace{s_{1}s'_{1}s_{1} \cdots}_{m} = \underbrace{s'_{1}s_{1}s'_{1}\cdots}_{m} $; $ s'_1s_1s_2s'_1s_1s_2 = s_2s'_1s_1s_2s'_1s_1 $; $ s_0s'_1s_0s'_1 = s'_1s_0s'_1s_0 $; $ s_0s_1s_0s_1 = s_1s_0s_1s_0 $; $ s_0s'_1s_1 = s'_1s_1s_0 $; $ \underbrace{s_{1}s_{0}s'_{1}s_{1}s'_{1}s_{1}\cdots}_{p+1} = \underbrace{s_{0}s'_{1}s_{1}s'_{1}s_{1}s'_{1}\cdots}_{p+1} $.

Let $ W $ be a Coxeter group and $ (S, P) $ be its presentation. Let $ J \subset S $ and $ W_J $ be a subgroup of $ W $ generated by $ J $. Then $ W_J $ is also a Coxeter group, which is called a parabolic subgroup of $ W $. A set of distinguished right coset representatives of $ W_J $ in $ W $ is defined in [3] as $ X_J : = \{w \in W | l(sw) > l(w) \forall s \in J\} $. Then for any $ w \in W $, it can be decomposed as $ w = vd $ with $ v \in W_J $ and $ d \in X_J $, and $ l(w) = l(v) + l(d) $. Assume $ (S, P) $ is a presentation of $ G(m, p, r) $, and let $ S' = S \setminus \{s_{r-1}\} $. The subgroup of $ G(m, p, r) $ generated by $ S' $ is denoted by $ G(m, p, r-1) $, which can also be thought of as a "parabolic" subgroup of $ G(m, p, r) $. In [4], Mac gave a set of complete right coset representatives of $ G(m, 1, r-1) $ in $ G(m, 1, r) $, which is denoted by $ X_r $. And she also proved that $ X_r $ is distinguished, according to which she can obtain a reduced expression for any element $ w \in G(m, 1, r) $ as $ w = d_1d_2 \cdots d_r $, where $ d_i \in X_i $ and $ G(m, 1, 0) $ is a trivial group.

We mean to give a set of distinguished right coset representatives of $ G(m, p, r-1) $ in $ G(m, p, r) $ when $ 1< p \leq m $, so that we are able to get a reduced expression for any element $ w \in G(m, p, r) $, like what Mac did in [4]. Well, it turns out to be not very easy. But as the first step, we can at least determine a set of complete right coset representatives of $ G(m, p, r-1) $ in $ G(m, p, r) $ (here $ r > 2 $), which is the main result of this paper.

Note that from now on, we always assume $ 1< p \leq m $ when $ G(m, p, r) $ is cited except special explanation.

Lemma 2.1 We have $ s_{1}s_{0} = s_{0}s_{1}'(s_{1}s_{1}')^{p-1} $ in $ G(m, p, r) $ when $ 1<p<m $.

Proof By the presentation of $ G(m, p, r) $ when $ 1<p<m $, we have relation

$ \underbrace{s_{1}s_{0}s_{1}'s_{1}s_{1}'s_{1}\cdots}_{p+1} = \underbrace{s_{0}s_{1}'s_{1}s_{1}'s_{1}s_{1}'\cdots}_{p+1}. $

If $ p $ is odd, this relation is

$ s_{1}s_{0}\underbrace{s_{1}'s_{1}\cdots s_{1}'s_{1}}_{p-1} = s_{0}s_{1}'\underbrace{s_{1}s_{1}'\cdots s_{1}s_{1}'}_{p-1}. $

So we have $ s_{1}s_{0} = s_{0}s_{1}'(s_{1}s_{1}')^{p-1} $.

If $ p $ is even, this relation is

$ s_{1}s_{0}s_{1}'\underbrace{s_{1}s_{1}'\cdots s_{1}s_{1}'}_{p-2} = s_{0}\underbrace{s_{1}'s_{1}\cdots s_{1}'s_{1}}_{p}. $

So we also have $ s_{1}s_{0} = s_{0}s_{1}'(s_{1}s_{1}')^{p-1} $.

Lemma 2.2 we have $ s_{2}s_{1}s_{1}'s_{2}(s_{1}'s_{1})^{k} = (s_{1}'s_{1})^{k}s_{2}s_{1}s_{1}'s_{2} $ for $ 1 \leq k \leq m $ in $ G(m, p, r) $.

Proof We prove by induction on $ k $. When $ k = 1 $, by the presentation of $ G(m, p, r) $, we have relation $ s_{2}s_{1}'s_{1}s_{2}s_{1}'s_{1} = s_{1}'s_{1}s_{2}s_{1}'s_{1}s_{2} $. So

$ \begin{eqnarray*} s_{2}s_{1}s_{1}'s_{2}s_{1}'s_{1} = s_{2}s_{1}s_{1}'s_{1}'s_{1}s_{2}s_{1}'s_{1}s_{2}s_{1}s_{1}'s_{2} = s_{1}'s_{1}s_{2}s_{1}s_{1}'s_{2}. \end{eqnarray*} $

Assume the conclusion is true for $ k = l $, i.e., we have $ s_{2}s_{1}s_{1}'s_{2}(s_{1}'s_{1})^{l} = (s_{1}'s_{1})^{l}s_{2}s_{1}s_{1}'s_{2} $. For $ k = l+1 $, we have

$ \begin{eqnarray*} s_{2}s_{1}s_{1}'s_{2}(s_{1}'s_{1})^{l+1} & = &s_{2}s_{1}s_{1}'s_{2}(s_{1}'s_{1})^{l}s_{1}'s_{1} = (s_{1}'s_{1})^{l}s_{2}s_{1}s_{1}'s_{2}s_{1}'s_{1}\\ & = &(s_{1}'s_{1})^{l}s_{1}'s_{1}s_{2}s_{1}s_{1}'s_{2} = (s_{1}'s_{1})^{l+1}s_{2}s_{1}s_{1}'s_{2}. \end{eqnarray*} $

Lemma 2.3 we have $ s_{2}(s_{1}s_{1}')^{k}s_{2}s_{1}' = s_{1}(s_{1}'s_{1})^{k-1}s_{2}(s_{1}s_{1}')^{k}s_{2} $ for $ 1 \leq k \leq m $ in $ G(m, p, r) $.

Proof We prove by induction on $ k $. When $ k = 1 $, since $ s_{1}'s_{2}s_{1}' = s_{2}s_{1}'s_{2} $ and $ s_{2}s_{1}s_{2} = s_{1}s_{2}s_{1} $, we have $ s_{2}s_{1}s_{1}'s_{2}s_{1}' = s_{2}s_{1}s_{2}s_{1}'s_{2} = s_{1}s_{2}s_{1}s_{1}'s_{2} $. Assume the conclusion is true for $ k = l $, i.e., we have $ s_{2}(s_{1}s_{1}')^{l}s_{2}s_{1}' = s_{1}(s_{1}'s_{1})^{l-1}s_{2}(s_{1}s_{1}')^{l}s_{2} $. For $ k = l +1 $, we have

$ \begin{eqnarray*} s_{2}(s_{1}s_{1}')^{l+1}s_{2}s_{1}' & = &s_{2}s_{1}s_{1}'(s_{1}s_{1}')^{l}s_{2}s_{1}' = s_{2}s_{1}s_{1}'s_{2}s_{1}(s_{1}'s_{1})^{l-1}s_{2}(s_{1}s_{1}')^{l}s_{2}\\ & = &s_{1}s_{2}s_{1}s_{1}'s_{2}s_{1}'s_{1}(s_{1}'s_{1})^{l-1}s_{2}(s_{1}s_{1}')^{l}s_{2} = s_{1}s_{2}s_{1}s_{1}'s_{2}(s_{1}'s_{1})^{l}s_{2}(s_{1}s_{1}')^{l}s_{2}. \end{eqnarray*} $

By Lemma 2.2, the last relation equals $ s_{1}(s_{1}'s_{1})^{l}s_{2}s_{1}s_{1}'s_{2}s_{2}(s_{1}s_{1}')^{l}s_{2} = s_{1}(s_{1}'s_{1})^{l}s_{2}(s_{1}s_{1}')^{l+1}s_{2} $.

Lemma 2.4 we have $ s_{2}(s_{1}s_{1}')^{k}s_{2}s_{1} = s_{1}(s_{1}'s_{1})^{k}s_{2}(s_{1}s_{1}')^{k}s_{2} $ for for $ 1 \leq k \leq m $ in $ G(m, p, r) $.

Proof We prove by induction on $ k $. When $ k = 1 $, since $ s_{2}s_{1}s_{1}'s_{2}s_{1}s_{1}' = s_{1}s_{1}'s_{2}s_{1}s_{1}'s_{2} $ and $ s_{1}'s_{2}s_{1}' = s_{2}s_{1}'s_{2} $, we have

$ \begin{eqnarray*} s_{2}s_{1}s_{1}'s_{2}s_{1} & = &s_{1}s_{1}'s_{2}s_{1}s_{1}'s_{2}s_{1}'s_{1}s_{2}s_{2}s_{1} = s_{1}s_{1}'s_{2}s_{1}s_{1}'s_{2}s_{1}'\\ & = &s_{1}s_{1}'s_{2}s_{1}s_{2}s_{1}'s_{2} = s_{1}s_{1}'s_{1}s_{2}s_{1}s_{1}'s_{2}. \end{eqnarray*} $

Assume the conclusion is true for $ k = l $, i.e., we have $ s_{2}(s_{1}s_{1}')^{l}s_{2}s_{1} = s_{1}(s_{1}'s_{1})^{l}s_{2}(s_{1}s_{1}')^{l}s_{2} $. For $ k = l+1 $, we have

$ \begin{eqnarray*} s_{2}(s_{1}s_{1}')^{l+1}s_{2}s_{1} & = &s_{2}s_{1}s_{1}'(s_{1}s_{1}')^{l}s_{2}s_{1} = s_{2}s_{1}s_{1}'s_{2}s_{1}(s_{1}'s_{1})^{l}s_{2}(s_{1}s_{1}')^{l}s_{2}\\ & = &s_{1}s_{1}'s_{1}s_{2}s_{1}s'_1s_2(s_{1}'s_{1})^{l}s_{2}(s_{1}s_{1}')^{l}s_{2} = s_{1}s_{1}'s_{1}(s_{1}'s_{1})^{l}s_{2}s_{1}s'_1s_2s_{2}(s_{1}s_{1}')^{l}s_{2}\\ & = &s_{1}(s_{1}'s_{1})^{l+1}s_{2}(s_{1}s_{1}')^{l+1}s_{2}. \end{eqnarray*} $

Note that the fourth equation holds by Lemma 2.2.

Theorem 2.5 Assume $ r>2 $. Let $ D^r_i = \{s_{r-1}s_{r-2} \cdots s_{2}(s_{1}s_{1}')^{i}, s_{r-1}s_{r-2} \cdots s_{2}(s_{1}s_{1}')^{i}s_{1}, s_{r-1}s_{r-2} \cdots s_{2}(s_{1}s_{1}')^{i}s_{2}, s_{r-1}s_{r-2} \cdots s_{2}(s_{1}s_{1}')^{i}s_{2}s_{3}, \ldots, s_{r-1}s_{r-2} \cdots s_{2}(s_{1}s_{1}')^{i}s_{2}s_{3} \ldots s_{r-1} \} $. Let $ D^r = \cup_{i = 0}^{m-1}D^r_{i} $, then $ D^r $ is a set of complete right coset representatives of $ G(m, p, r-1) $ in $ G(m, p, r) $ when $ 1 < p \leq m $.

Proof Let $ W = G(m, p, r) $ and $ L = G(m, p, r-1) $. We want to show $ W = \bigcup_{d \in D^r}Ld $. It's obvious that $ \bigcup_{d \in D^r}Ld \subset W $ and $ |L||D^r| = |W| $, so we only need to show that $ \forall s \in S = \{s_{0}, s_{1}', s_{1}, \ldots, s_{r-1}\} $, and $ \forall d \in D^r $, there exists $ d' \in D^r $ such that $ Lds = Ld' $. We discuss in the following cases.

(a) Assume $ s = s_0 $, note that this case happens only when $ 1<p<m $. We have the relation $ s_{1}'s_{1}s_{0} = s_0s_{1}'s_{1} $ and $ s_0s_j = s_js_0 $ for $ j>1 $.

(a.1) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i} $,

$ \begin{eqnarray*} ds_{0} = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{m-i}s_{0} = s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{m-i} = s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i} \in Ld. \end{eqnarray*} $

(a.2) If $ d = s_{r-1}s_{r-2} \ldots s_{2}(s_{1}s_{1}')^{i}s_{1} $, by lemma 2.1 we have $ s_{1}s_{0} = s_{0}s_{1}'(s_{1}s_{1}')^{p-1} $, then

$ \begin{eqnarray*} ds_{0} & = &s_{r-1}s_{r-2}\ldots s_{2}s_{1}(s_{1}'s_{1})^{i}s_{0} = s_{r-1}s_{r-2}\ldots s_{2}s_{1}s_{0}(s_{1}'s_{1})^{i}\\ & = &s_{r-1}s_{r-2}\ldots s_{2}s_{0}s_{1}'(s_{1}s_{1}')^{p-1}(s_{1}'s_{1})^{i} = s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{p-1}s_{1}'(s_{1}'s_{1})^{i}\\ & = &s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{p-1}(s_{1}s_{1}')^{i-1}s_1. \end{eqnarray*} $

The last relation equals

$ s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i-p}s_1 \in Ld' $

with $ d' = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i-p}s_1 $ when $ p \leq i $; or equals

$ s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{m-(p-i)}s_1 \in Ld' $

with $ d' = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{m-(p-i)}s_1 $ when $ i<p $.

(a.3) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j}\; (j \geq 2) $, then

$ \begin{eqnarray*} ds_{0} & = &s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{m-i}s_{0}s_{2}\ldots s_{j} = s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}'s_{1})^{m-i}s_{2}\ldots s_{j}\\ & = &s_{0}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j} \in Ld. \end{eqnarray*} $

(b) Assume $ s = s_{1}' $,

(b.1) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i} $, then

$ ds_{1}' = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1}' = \begin{cases} s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i-1}s_{1} \in D^r_{i-1} , \text {for}\, i>0, \\ s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{m-1}s_{1} \in D^r_{m-1} , \text {for}\, i = 0. \end{cases} $

(b.2) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1} $, then

$ ds_{1}' = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1}s_{1}' = \begin{cases} s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i+1} \in D^r_{i+1} , \text{ for}\, i < m-1, \\ s_{r-1}s_{r-2}\ldots s_{2} \in D^r_{0} , \text {for}\, i = m-1. \end{cases} $

(b.3) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j}\; (j \geq 2) $, then

$ ds_{1}' = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}s_{1}'s_{3}\ldots s_{j}. $

When $ i = 0 $, $ ds_{1}' = s_{1}'d \in Ld $; when $ i>0 $, by Lemma 2.3, we have $ s_{2}(s_{1}s_{1}')^{i}s_{2}s_{1}' = s_{1}(s_{1}'s_{1})^{i-1}s_{2}(s_{1}s_{1}')^{i}s_{2}. $ Then

$ \begin{eqnarray*} ds_{1}' & = &s_{1}(s_{1}'s_{1})^{i-1}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}s_{3}\ldots s_{j} \in Ld. \end{eqnarray*} $

(c) Assume $ s = s_{j}, 1 \leq j \leq r-1 $.

(c.1) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i} $.

(c.1.1) When $ j = 1 $ or $ 2 $, $ ds_j \in D^r_i $.

(c.1.2) When $ j \geq 3 $, $ ds_j = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{j} = s_{j-1}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i} \in Ld $.

(c.2) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1} $.

(c.2.1) When $ j = 1 $, $ ds_j \in D^r_i $.

(c.2.2) When $ j = 2 $, $ ds_{j} = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1}s_{2} $. By Lemma 2.4, we have

$ s_{2}(s_{1}s_{1}')^{i}s_{1}s_{2} = s_{1}(s_{1}'s_{1})^{i}s_{2}(s_{1}s_{1}')^{i}s_{1}. $

Then

$ ds_{j} = s_{1}(s_{1}'s_{1})^{i}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1} \in Ld. $

(c.2.3) When $ j \geq 3 $, $ ds_j = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1}s_{j} = s_{j-1}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{1}\in Ld. $

(c.3) If $ d = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k}\; (2 \leq k \leq r-1) $.

(c.3.1) When $ j = 1 $, $ ds_{j} = s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}s_{1}\ldots s_{k} $. By Lemma 2.4, we have

$ s_{2}(s_{1}s_{1}')^{i}s_{1}s_{2} = s_{1}(s_{1}'s_{1})^{i}s_{2}(s_{1}s_{1}')^{i}s_{1}. $

Then $ ds_j = s_{1}(s_{1}'s_{1})^{i}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k} \in Ld $.

(c.3.2) When $ 2 \leq j \leq k-1 $,

$ \begin{eqnarray*} ds_{j} & = &s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j-1}s_{j}s_{j+1}\ldots s_{k}s_{j}\\ & = &s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j-1}s_{j}s_{j+1}s_{j}\ldots s_{k}\\ & = &s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{j-1}s_{j+1}s_{j}s_{j+1}\ldots s_{k}\\ & = &s_{r-1}s_{r-2}\ldots s_{j+1}s_{j}s_{j+1}s_{j-1}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2} \ldots s_{k}\\ & = &s_{r-1}s_{r-2}\ldots s_{j}s_{j+1}s_{j}s_{j-1}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2} \ldots s_{k}\\ & = &s_{j}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k} \in Ld. \end{eqnarray*} $

(c.3.3) When $ j = k $ or $ k+1 $, $ ds_j \in D^r_i $.

(c.3.4) When $ k+2 \leq j \leq r-1 $,

$ \begin{eqnarray*} ds_{j} & = &s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k}s_{j}\\ & = &s_{r-1}s_{r-2}\ldots s_{j+1}s_{j}s_{j-1}s_{j}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k}\\ & = &s_{r-1}s_{r-2}\ldots s_{j+1}s_{j-1}s_{j}s_{j-1}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k}\\ & = &s_{j}s_{r-1}s_{r-2}\ldots s_{2}(s_{1}s_{1}')^{i}s_{2}\ldots s_{k} \in Ld \end{eqnarray*} $

Up to now, we have discussed all the cases, so the theorem follows.