A class of fuzzy subgroups of finite reflection groups

Abstract

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ. We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ.

Keywords

Preferential equality fuzzy subgroup reflection group subgroup parabolic fuzzy subgroup dihedral group

1 Introduction

Das [2] studied fuzzy subgroups by introducing an equivalence relation based on α-cuts, on the set of all fuzzy subgroups of a fixed group. This idea was developed further to study a generalization of equality of crisp subgroups. We call such an equivalence relation a preferential equality and an equivalence class of fuzzy subgroups, a preferential fuzzy subgroup, see for instance [6] and others, [4] and [9]. The importance of the study of parabolic subgroups of reflection groups cannot be over emphasized from a geometric and algebraic point of view, see for instance [3]. Also it may be desirable to look at certain reflections as being more natural than others since the geometric picture that emerges from applying the reflections in a particular sequence is esthetically more natural. see for instance Example 2.1. Hence in this paper we use the idea of preferential fuzzy subgroups to characterize certain fuzzy subgroups of finite reflection groups generated by simple reflections. We call these objects preferential fuzzy parabolic subgroups and they are in one-to-one correspondence with subsets of simple root system Δ indexing a set of simple reflections.

In Section 2, we recall some well-known facts regarding finite reflection groups, simple reflections, root systems and parabolic subgroups. In Section 3 we collect results on fuzzy subgroups such as α-cuts, preferential equality, preferential fuzzy subgroups, support and core that are essential. In Section 4 we develop ideas of fuzzy subgroups of finite reflection groups introducing notions of preferential fuzzy parabolic subgroups and characterize such objects within the context of preferential equality. Throughout we have used the example of $D_{4}$ to illustrate the ideas of the paper.

2 Finite reflection groups

All the material in this section is a summary of general results on reflection groups and mostly taken from the book “Reflection groups and Coxeter groups” by Humphreys [3].

2.1 Basic concepts

As parabolic subgroups are certain subgroups of finite reflection groups we begin with an abstract definition of a finite reflection group. Let V be a finite dimensional Euclidean space of dimension n and O (V) be the orthogonal group of V consisting of all orthogonal transformations of V, with the identity transformation denoted by e. We recall that a linear transformation R on V is called a reflection if R fixes a hyperplane H pointwise and sends a vector v orthogonal to H to its negative -v, that is, R (v) = - v and hence R (L_v) = L_v where L_v is the line aligned with vector v. In this case, the reflection R is denoted by R_v and is called the reflection generated by v. We note that R_v is an element of order 2 in O (V) and may be described by a basic formula using an inner-product (u, v) defined on V. The image of u ∈ V under R_v is given by $R_{v} (u) = u - \frac{2 (u, v)}{(v, v)} v$ . A finite subgroup W of O (V) generated by a finite number of reflections is called a finite reflection group. We use w for a general element of W. Some examples of finite reflection groups are the classical dihedral groups $D_{n}$ of order 2n, the symmetric groups $S_{n}$ regarded as a finite subgroup of O (V)generated by transpositions, groups B_n arising as semi-direct products of symmetric groups $S_{n}$ and the group of sign changes represented by $(ℤ / 2 ℤ)^{n}$ and lastly the subgroups D_n of B_n that are of index 2. Obviously finite reflection groups arise in geometric contexts and have been studied extensively, see for instance [3] and [1].

2.2 Root systems of reflection groups

We now briefly recall what are roots and root systems that are associated with finite reflection groups. These concepts are standard in Weyl group theory, in particular finite reflection groups. See, for instance [3] and [1].

We call a finite group W, a finite reflection group if it is generated by a set of reflections. Let $L$ be a finite collection of non-zero vectors v ∈ V such that $R_{v} (L) = L$ and $L_{v} \cap L = {\pm v}$ . Without loss of generality, by normalization, we can assume the vectors in $L$ to be unit vectors. If W is generated by all reflections R_v, where v is an element of $L$ , then $L$ is called a root system associated with W or simply a root system for W. In this case any element belonging to $L$ is called a root of W. We call a subset Π of $L$ a positive (or negative) system if it contains all the roots which are positive (or negative) relative to some total ordering of V. For instance the set of four vectors {(1, 0) , (1, 1) , (0, 1) , (-1, 1)} in $ℝ^{2}$ is a positive system for $D_{4}$ while the corresponding negative vectors form a negative system. In general it is clear that the positive and negative system for a W come in disjoint pair. A subset Δ of $L$ is called a simple root system if it is a basis for the span of $L$ and if each $α \in L$ is a linear combination of elements of Δ with all the coefficients of the same sign. Further a reflection associated with an element of Δ is called a simple reflection. Three facts are well-known in the literature:

A system of simple reflections exists for a given reflection group

Any two simple systems Δ and Δ′ are conjugate in W in the sense that every β ∈ Δ′ is of the form wα for some α ∈ Δ, and

Simple reflections associated with a simple root system generate W. The cardinality of any simple root system is called the rank of W.

2.3 Generation of W

In Subsection 2.1, we discussed how W is generated by a set S of all simple reflections associated with a simple system Δ. We now describe the generation of W using relations on simple reflections. For any two simple reflections s_α, s_β ∈ S for α, β ∈ Δ, let $m (α, β) \in ℕ$ be the order of s_α . s_β in W. Then $(s_{α} . s_{β})^{m (α, β)} = e for α, β \in Δ .$

So W can be viewed as generated by all the elements of the set S with the above specified relations. For instance, when α = β, and from the observation $(s_{α}^{2}) = e$ , we can conclude that m (α, α) =1 since (s_α . s_β) ^m(α,β) = e becomes (s_α . s_α) ^m(α,α) = e implying ${(s_{α}^{2})}^{m (α, α)} = e .$

Example 2.1. Let $W = D_{4}$ . The vectors α and β are shown below with respect to the usual xy-plane forming a simple system Δ = {α, β} for W. Let s_α, s_β ∈ S be the corresponding simple reflections generating W. We note that both s_α . s_β and s_β . s_α are of order 4, (m (α, β) =4 and m (β, α) =4)), but one a clockwise rotation and the other is anti-clockwise rotation.

In certain practical situations it may be useful to consider the reflection s_α to be more suitable than s_β or vice-versa. That will be the basis for considering preferential fuzzy parabolic subgroups.

In the literature any group W generated by a system of generators satisfying relations as described above is called a Coxeter group and the pair (W, S) is called a Coxeter system. Thus a finite reflection group is a typical example of a Coxeter group. An illuminating discussion of this example is found in [8].

2.4 Parabolic subgroups of W

As in the previous Subsection 2.3, let Δ be a simple system and S = {s_α : α ∈ Δ} be the corresponding set of simple reflections in W.

Definition 2.2. For any subset I of S, the subgroup W_I of W generated by all simple reflections in I is called a parabolic subgroup of W.

The two trivial parabolic subgroups are W_∅ = {e} and W_S = W. The effect of replacing one simple system Δ by another Δ′ on the parabolic subgroups associated with Δ and Δ′ is one of conjugacy, that is, if Δ′ = wΔ for some w ∈ W, then the parabolic subgroup W_I is replaced by wW_Iw^-1. It is clear not every subgroup of a finite reflection group is parabolic. A fortiori a non-parabolic subgroup remains the same under any simple root system. For instance a rotation subgroup of order 4 in $D_{4}$ is non-parabolic under any simple root system for $D_{4}$ .

We now think of the power set $P (S)$ of all subsets of S as a lattice under the usual union ∨ and intersection ∧ of subsets of S. Similarly the set of all parabolic subgroups $W (S)$ of W associated with S, can be considered as a lattice with union ∨ and the intersection ∧ of two parabolic subgroups defined by W_I ∨ W_J is the subgroup generated by W_I and W_J in W and W_I ∧ W_J is simply the set intersection of the subgroups W_I and W_J. Hence we have thefollowing

Proposition 2.3. (Humphreys [3]) Under the correspondence I ↔ W_I, the lattice $W (S)$ of all parabolic subgroups {W_I : I ⊂ S} is lattice isomorphic to the power set $P (S)$ of all subsets of S under the union (∨) and intersection (∧) as lattice operations.

3 Fuzzy subgroups

1° Summary of basic concepts.

We use I = [0, 1], the real unit interval as a chain with the usual ordering in which ∧ stands for infimum (inf) (or intersection) and ∨ stands for supremum (sup)(or union). A fuzzy subset of a set W is a mapping μ : W → I. The union, intersection of two fuzzy sets, and complementation of a fuzzy set are defined using sup and inf pointwise, and 1 - μ operator pointwise, respectively. We denote the set of all fuzzy subsets of W by I^W. Throughout this paper we take W to be a finite reflection group and W_φ to be the trivial subgroup {1} denoting the identity of W. By an t-cut of μ for a real number t in I, we mean a subset μ^t = {w ∈ W : μ (w) ≥ t} of W. A fuzzy set μ is said to be a fuzzy subgroup if μ (w₁ w₂) ≥ μ (w₁) ∧ μ (w₂) forall w₁, w₂ ∈ W and μ (w) = μ (w^-1) for all w ∈ W, see for instance [7]. In this paper we assume μ (1) =1. From this assumption we notice that the only admissible fuzzy subgroup of the trivial group is μ (1) = 1. By core and support of μ we mean the crisp subsets of W given by core (μ) = {w ∈ W : μ (w) =1} and supp (μ) = {w ∈ W : μ (w) ≠0}. Clearly the t-cut of a fuzzy subgroup μ is a subgroup.

2° Preferential equality of fuzzy subgroups.

We recall from [6] the notion of preferential equality of two fuzzy subgroups μ and ν of W. This notion is an equivalence relation on I^W, written as μ ∼ ν and defined by the

Definition 3.1. Two fuzzy subgroups μ ∼ ν if and only if the following two conditions are satisfied: for all w₁, w₂ ∈ W,

$\begin{matrix} (i) μ (w_{1}) > μ (w_{2}) \Leftrightarrow ν (w_{1}) > ν (w_{2}), \\ (ii) μ (w) = 0 \Leftrightarrow ν (w) = 0 . \end{matrix}$ (3.1)

As a contra-definition we say two fuzzy subgroups are preferentially distinct if they are not preferentially equal. Thus two fuzzy subgroups are either preferentially equal or preferentially distinct. A further remark is useful and the proof is found in the samepaper [6].

Remark 3.2. If two fuzzy subgroups are preferentially equal then they must have the same number of membership values in I, that is, |Im (μ) | = |Im (ν) |. In contrast even if Im (μ) = Im (ν), μ and ν may be preferentially distinct. The equivalence class containing μ is called a preferential fuzzy subgroup and is denoted by $\bar{μ}$ .

4 Fuzzy parabolic subgroups

4.1 Definition and basic properties

We tacitly assume unless otherwise stated that W is a finite reflection group and Δ is a simple root system in V. By specializing the general definition of a fuzzy subgroup of a group, we define a fuzzy parabolic subgroup of a finite reflection group W associated with a simple root system Δ in the following way.

Definition 4.1. A fuzzy parabolic subgroup of W is a fuzzy subgroup μ : W → I such the t-cut μ^t for every t ∈ I is a parabolic subgroup associated with a specified Δ in V.

We refer to Example 2.1 with $W = D_{4}$ . Then it is easy to check that the fuzzy subset μ : W → I defined by $μ (x) = {\begin{matrix} 1, & x = e \\ \frac{1}{2}, & x = s_{α} \\ \frac{1}{3}, & x = s_{β} \end{matrix}$ (4.1) on simple reflections s_α and s_β is a fuzzy parabolic subgroup of W. We note that not all fuzzy subgroups of W need be fuzzy parabolic subgroups of W since not all subgroups of W need be parabolic in the crisp case. For instance the fuzzy subgroup μ defined by $μ (e) = 1, μ (s_{α} . s_{β})^{2}) = \frac{1}{2}$ is a fuzzy subgroup of W but not a fuzzy parabolic subgroup of W. However preferential equality on fuzzy subgroups restricted to the set of fuzzy parabolic subgroups does not lead to any unexpected complication. That is, if one of the two preferentially equal fuzzy subgroups is fuzzy parabolic then the other must also be fuzzy parabolic which we state as the following

Proposition 4.2. Suppose μ is a fuzzy parabolic subgroup of W and ν is a fuzzy subgroup of W that is preferentially equal to μ. Then ν is also a fuzzy parabolic subgroup of W.

Proof. Suppose μ ∼ ν. Clearly the supports and cores of μ and ν must be the same. Further, suppose t₀ = 1, t₁, t₂, ⋯ , t_m = 0 and $t_{0}^{'} = 1, t_{1}^{'}, t_{2}^{'}, \dots, t_{n}^{'} = 0$ are the membership values of μ and ν respectively in the decreasing order, then m = n by the Remark 3.2 just after Definition 3.1. Also each segment [t_i-1, t_i] is equipotent to the segment $[t_{i - 1}^{'}, t_{i}^{'}]$ for i = 1, 2, ⋯ , m = n, by some bijective function f on I which takes t_i to $t_{i}^{'}$ for i = 0, 1, 2 ⋯ , m = n. In this case, take t′ = f (t) for any given t in I. Thus for each t there exists a t′ such that μ^t = ν^t′. Thus every t-cut of ν is a parabolic subgroup implying that ν is a fuzzy parabolic subgroup of W. □

The above proposition leads to a preferential fuzzy parabolic subgroup of W as an equivalence class of fuzzy parabolic subgroups μ : W → I, associated with a specified Δ in V, under the equivalence of preferential equality of fuzzy subgroups. This definition is unambiguous since the Proposition 4.2 assures that any fuzzy subgroup that is preferentially equal to a fuzzy parabolic subgroup is itself a fuzzy parabolic subgroup. Thus one can talk of a preferential fuzzy parabolic subgroup in general without any danger of ambiguity.

4.2 The correspondence

In this section we set up a one-to-one correspondence between the class of preferential fuzzy parabolic subgroups of W with respect to a fixed simple root system Δ with its associated set S of simple reflections and the class of preferential fuzzy subsets of S. First we briefly sketch how to generate a fuzzy parabolic subgroup from a given a fuzzy subset of the class S of simple reflections associated with a fixed simple root system Δ.

Clearly if I₁ and I₂ are two subsets of S of simple reflections indexed by Δ such that I₁ ⊆ I₂, then the parabolic subgroups W_{I
₁} and W_{I
₂} of W generated by the sets I₁ and I₂ respectively satisfy the natural inclusion W_{I
₁} ⊆ W_{I
₂} as subgroups of W. This simple observation allows us to define a fuzzy parabolic subgroup W_μ : W → I of W as follows.

Construction 4.3. Let μ : S → I be a fuzzy subset of S = {s_α : α ∈ Δ} defined by μ (s_α) = t₀ if s_α ∈ I₀ and $μ (s_{α}) = t_{i} if s_{α} \in I_{i} \ I_{i - 1} for i = 1, \dots, n$ where I₀ ⊆ I₁ ⊆ I₂ ⊆ ⋯ ⊆ I_n-1 ⊆ I_n = S and 1 ≥ t₁ ≥ t₂ ≥ ⋯ ≥ t_n ≥ 0. The fuzzy subset μ can be written as

μ = ∨ {t_iχ_{I
_i} : 0 ≤ i ≤ n} where χ stands for the characteristic function of a subset.

Let <I_i> be the subgroup generated by the set I_i in W for i = 0, 1, ⋯ n - 1. Now we define another fuzzy subset W_μ of W corresponding to μ as follows: W_μ (w) =1 if w∈ < I₀ > and for i = 1, 2, ⋯ , n $W_{μ} (w) = t_{i} ifw \in < I_{i} > \ < I_{i - 1} >$ (4.2) where <I₀ > ⊆ < I₁ > ⊆ < I₂ > ⊆ ⋯ ⊆ < I_n-1 > ⊆ W and 1 ≥ t₁ ≥ t₂ ≥ ⋯ ≥ t_n ≥ 0. W_μ = ∨ {t_iχ_{<I_i>} : 0 ≤ i ≤ n}.

It is easy to check that W_μ is a fuzzy parabolic subgroup of W.

With above construction we have the following

Theorem 4.4. Suppose μ and ν are two fuzzy subsets of S. Let W_μ and W_ν be the corresponding fuzzy parabolic subgroups of W with respect to a fixed simple root system Δ. Then μ ∼ ν in S if and only if W_μ ∼ W_ν in W.

Proof. Using the Construction 4.2 above, μ and ν can be represented by μ = ∨ {t_iχ_{I
_i} : 0 ≤ i ≤ n} and ν = ∨ {r_iχ_{J
_i} : 0 ≤ i ≤ n} for suitable values of t_i’s and r_i’s in the unit interval I. It is easy to check that μ ∼ ν in S if

t_i > t_i′ ⇔ r_i > r_i′ I_i = J_i, and t_i = 0 ⇔ r_i = 0, for 1 ≤ i, i′ ≤ n .

See [5]. From (ii) above we get I_i = J_i if and only if <I_i> = < J_i > in W. Hence μ ∼ ν in S if and only if W_μ ∼ W_ν in W. □

5 Conclusion

In conclusion this paper sketches a framework of how to study the fuzzy subgroups of reflection groups under preferential equality. Elements of reflection groups are linear transformations on vector spaces. Therefore the above results can be considered in the context of reflections as linear transformations of fuzzy vector spaces. Further studies are needed to work out how precisely linear transformations between fuzzy subspaces would lead to simple reflections and thereby leading to general reflections in fuzzy vector spaces.

The authors thank the National Research Foundation of South Africa (NRF) and the second author thanks RC of Rhodes University for financial support. We both thank Sithembele Nkonkobe for his contribution.

References

Benson

C.T.

, Grove

L.C.

Finite Reflection Groups, 2nd edition, Graduate Texts in Math, Springer, 1985.

Das

P.S.

, Fuzzy groups and level subgroups, J Math Anal and Appl 84 (1981), 264–269.

Humphreys

J.E.

Reflection and Coxeter groups, Cambridge University Press, 1992.

Jain

, Fuzzy subgroups and certain equivalence relations, Iran J Fuzzy Systems 3 (2006), 75–91.

Makamba

B.B.

, Murali

, Sum and quotient of preferential fuzzy subgroups, Far East J Math Sci, (FJMS) 69(2) (2012), 233–249.

Murali

, Makamba

B.B.

, On an equivalence of fuzzy subgroups I, Fuzzy Sets and Systems 123 (2001), 163–168.

Rosenfeld

, Fuzzy groups, J Math Anal Appl 35 (1971), 512–517.

Taylor

A Short Course in Lie Theory, University of Aberdeen, Scotland, UK, (2009) http://www.abdn.ac.uk/r01jmt8/

Zhang

, Zou

, A note on an equivalence relation on fuzzy subgroups, Fuzzy Sets and Systems 95 (1998), 243–247.