ON COSET DECOMPOSITIONS OF THE COMPLEX REFLECTION GROUPS G(M,P,R)

XU Jing-leiWANG Yan-jieWANG Li

(1.School of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China)

(2.The Affiliated High School of Shanghai University, Shanghai 200444, China )

(3.Changshu High School, Changshu 215516, China)

Abstract: We study the decomposition of the imprimitive complex reflection group G(m,p,r)into right coset, where m,p,r are positive integers, and p divides m.By use of the software GAP to compute some special cases when m,p,r are small integers, we deduce a set of complete right coset representatives of the parabolic subgroup G(m,p,r −1) in the group G(m,p,r) for general cases, which lays a foundation for further study the distinguished right coset representatives of G(m,p,r −1) in G(m,p,r).

Keywords: right coset representatives; imprimitive complex reflection groups

1 Introduction

Let N (respectively, Z, R, 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 R-subspace V0of V such that the canonical map C ⊗RV0→ 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 =V1⊕ V2⊕ ···⊕ Vtsuch that every element w ∈ G is a permutation on the set {V1,V2,··· ,Vt}.

For any m,p,r ∈ N with p | m (read “ p divides m ”), let G(m,p,r) be the group consisting of all r×r monomial matrices whose non-zero entries a1,a2,··· ,arare mth roots of unity withwhere a is in the i-th row of the monomial matrix.In [1],iShephard 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 ≤2 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

Let W be a Coxeter group and (S,P) be its presentation.Let J ⊂S and WJbe a subgroup of W generated by J.Then WJis also a Coxeter group,which is called a parabolic subgroup of W.A set of distinguished right coset representatives of WJin W is defined in[3]as XJ:= {w ∈ W|l(sw) > l(w)∀s ∈ J}.Then for any w ∈ W, it can be decomposed as w =vd with v ∈ WJand d ∈ XJ, and l(w)=l(v)+l(d).Assume (S,P) is a presentation of G(m,p,r), and let S= S{sr−1}.The subgroup of G(m,p,r) generated by Sis 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 Xr.And she also proved that Xris distinguished, according to which she can obtain a reduced expression for any element w ∈ G(m,1,r)as w =d1d2···dr,where di∈Xiand 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

2), which is the main result of this paper.

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

2 Main Results

Lemma 2.1We havein G(m,p,r) when 1

ProofBy the presentation of G(m,p,r) when 1

If p is odd, this relation is

If p is even, this relation is

Lemma 2.2we havefor 1 ≤ k ≤ m in G(m,p,r).

ProofWe prove by induction on k.When k = 1, by the presentation of G(m,p,r),we have relationSo

Assume the conclusion is true for k =l, i.e., we haveFor k =l+1, we have

Lemma 2.3we havefor 1 ≤ k ≤ m in G(m,p,r).

ProofWe prove by induction on k.When k =1, sinceand s2s1s2=s1s2s1, we haveAssume the conclusion is true for k =l, i.e., we haveFor k =l+1, we have

By Lemma 2.2,the last relation equals

Lemma 2.4we havefor for 1 ≤ k ≤ m in G(m,p,r).

ProofWe prove by induction on k.When k = 1, sincewe have

Assume the conclusion is true for k = l, i.e., we haveFor k =l+1, we have

Note that the fourth equation holds by Lemma 2.2.

ProofLet W =G(m,p,r) and L=G(m,p,r −1).We want to showIt’s obvious thatand |L||Dr|=|W|, so we only need to show that ∀s ∈ S =and ∀d ∈ Dr, there exists d∈ Drsuch that Lds=Ld.We discuss in the following cases.

(a) Assume s = s0, note that this case happens only when 1 < p < m.We have the relationand s0sj=sjs0for j >1.

The last relation equals

(c.3.3) When j =k or k+1, dsj∈.

(c.3.4) When k+2 ≤ j ≤ r − 1,

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