Coxeter group - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Coxeter group, named after H.S.M. Coxeter

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....

, is an abstract group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection group

Reflection group

In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a...

s; the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

s of regular polyhedra

Regular polyhedron

A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...

are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced as abstractions of reflection group

Reflection group

s, and finite Coxeter groups were classified in .

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

s of regular polytope

Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...

s, and the Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

s of simple Lie algebras. Examples of infinite Coxeter groups include the triangle group

Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...

s corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane

Hyperbolic space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

, and the Weyl groups of infinite-dimensional Kac–Moody algebra

Kac–Moody algebra

In mathematics, a Kac–Moody algebra is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix...

s.

Standard references include and .

Definition

Formally, a Coxeter group can be defined as a group

Group (mathematics)

with the presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

where

and

for

.
The condition

means no relation of the form

should be imposed.

The pair (W,S) where W is a Coxeter group with generators S={r₁,...,r_n} is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type BC₃ and A₁xA₃ are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).

A number of conclusions can be drawn immediately from the above definition.

The relation m_{i i} = 1 means that (r_ir_i )¹ = (r_i )² = 1 for all i ; the generators are involutions.
If m_{i j} = 2, then the generators r_i and r_j commute. This follows by observing that

xx = yy = 1,

together with

xyxy = 1

implies that

xy = x(xyxy)y = (xx)yx(yy) = yx.

Alternatively, since the generators are involutions,

, so

, and thus is equal to the commutator

Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...

In order to avoid redundancy among the relations, it is necessary to assume that m_{i j} = m_{j i}. This follows by observing that

yy = 1,

together with

(xy)^m = 1

implies that

(yx)^m = (yx)^myy = y(xy)^my = yy = 1.

Alternatively,

and

are conjugate elements, as

The Coxeter matrix is the n×n, symmetric matrix with entries m_{i j}. Indeed, every symmetric matrix with positive integer and ∞ entries and with 1's on the diagonal such that all nondiagonal entries are greater than 1 serves to define a Coxeter group.
The Coxeter matrix can be conveniently encoded by a Coxeter graph, as per the following rules.

The vertices of the graph are labelled by generator subscripts.
Vertices i and j are connected if and only if m_{i j} ≥ 3.
An edge is labelled with the value of m_{i j} whenever it is 4 or greater.

In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected component

Connected component (graph theory)

In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example, the graph shown in the illustration on the right has three connected components...

s, the associated group is the direct product

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of the groups associated to the individual components.
Thus the disjoint union

Disjoint union

In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...

of Coxeter graphs yields a direct product

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

of Coxeter groups.

An example

The graph in which vertices

Vertex (graph theory)

In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

1 through n are placed in a row with each vertex connected by an unlabelled edge to its immediate neighbors gives rise to the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

S_n+1; the generators

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

correspond to the transpositions (1 2), (2 3), ... (n n+1). Two non-consecutive transpositions always commute, while (k k+1) (k+1 k+2) gives the 3-cycle (k k+2 k+1). Of course this only shows that S_n+1 is a quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

of the Coxeter group described by the graph, but it is not too difficult to check that equality holds.

Connection with reflection groups

Coxeter groups are deeply connected with reflection group

Reflection group

s. Simply put, Coxeter groups are abstract groups (given via a presentation), while reflection groups are concrete groups (given as subgroups of linear groups or various generalizations). Coxeter groups grew out of the study of reflection groups — they are an abstraction: a reflection group is a subgroup of a linear group generated by reflections (which have order 2), while a Coxeter group is an abstract group generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certain form (

, corresponding to hyperplanes meeting at an angle of

, with

being of order k abstracting from a rotation by

).

The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group. For finite reflection groups, this yields an exact correspondence: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group may not admit a representation as a reflection group.

Historically, proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form

), and indeed this paper introduced the notion of a Coxeter group, while proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.

Finite Coxeter groups

Classification

The finite Coxeter groups were classified in , in terms of Coxeter–Dynkin diagrams; they are all represented by reflection group

Reflection group

s of finite-dimensional Euclidean spaces.

The finite Coxeter groups consist of three one-parameter families of increasing rank

one one-parameter family of dimension two,

and six exceptional

Exceptional object

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects as well as a finite number of exceptions that don't fit into any series. These are known as exceptional...

groups:

and

Weyl groups

Many, but not all of these, are Weyl groups, and every Weyl group

Weyl group

can be realized as a Coxeter group. The Weyl groups are the families

and

and the exceptions

and

denoted in Weyl group notation as

The non-Weyl groups are the exceptions

and

and the family

except where this coincides with one of the Weyl groups (namely

and

).

This can be proven by comparing the restrictions on (undirected) Dynkin diagrams with the restrictions on Coxeter diagrams of finite groups: formally, the Coxeter graph can be obtained from the Dynkin diagram by discarding the direction of the edges, and replacing every double edge with an edge labelled 4 and every triple edge by an edge labelled 6. Also note that every finitely generated Coxeter group is an Automatic group

Automatic group

In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.More...

. Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above. Geometrically, this corresponds to the crystallographic restriction theorem

Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold...

, and the fact that excluded polytopes do not fill space or tile the plane – for

the dodecahedron (dually, icosahedron) does not fill space; for

the 120-cell (dually, 600-cell) does not fill space; for

a p-gon does not tile the plane except for

or 6 (the triangular, square, and hexagonal tilings, respectively).

Note further that the (directed) Dynkin diagrams B_n and C_n give rise to the same Weyl group (hence Coxeter group), because they differ as directed graphs, but agree as undirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but having the same symmetry group.

Properties

Some properties of the finite Coxeter groups are given in the following table:

Group symbol	Alternate symbol	Bracket notation Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...	Rank	Order Order (group theory) In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....	Related polytopes Uniform polytope A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
A_n	A_n	[3ⁿ]	n	(n + 1)!	n-simplex Simplex In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...	..
BC_n	C_n	[4,3^n-1]	n	2ⁿ n!	n-hypercube Hypercube In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An... / n-cross-polytope Cross-polytope In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...	...
D_n	B_n	[3^n-3,1,1]	n	2ⁿ⁻¹ n!	n-demihypercube	...
E₆	E₆	[3^2,2,1]	6	72x6! = 51840	2₂₁, 1₂₂
E₇	E₇	[3^3,2,1]	7	72x8! = 2903040	3₂₁, 2₃₁, 1₃₂
E₈ E8 (mathematics) In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...	E₈	[3^4,2,1]	8	192x10! = 696729600	4₂₁, 2₄₁, 1₄₂
F₄	F₄	[3,4,3]	4	1152	24-cell
G₂	-	[6]	2	12	hexagon
H₂	G₂	[5]	2	10	pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
H₃	G₃	[3,5]	3	120	icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... / dodecahedron
H₄	G₄	[3,3,5]	4	14400	120-cell / 600-cell
I₂(p)	D₂^p	[p]	2	2p	p-gon Regular polygon A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

Symmetry groups of regular polytopes

All symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

s of regular polytope

Regular polytope

s are finite Coxeter groups. Note that dual polytopes have the same symmetry group.

There are three series of regular polytopes in all dimensions. The symmetry group of a regular n-simplex

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

is the symmetric group

Symmetric group

S_n+1, also known as the Coxeter group of type A_n. The symmetry group of the n-cube

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

and its dual, the n-cross-polytope

Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...

is BC_n, and is known as the hyperoctahedral group

Hyperoctahedral group

In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. Groups of this type are identified by a parameter n, the dimension of the hypercube....

.

The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups. In two dimensions, the dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

s, which are the symmetry groups of regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s, form the series I₂(p). In three dimensions, the symmetry group of the regular dodecahedron and its dual, the regular icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, is H₃, known as the full icosahedral group. In four dimensions, there are three special regular polytopes, the 24-cell, the 120-cell, and the 600-cell. The first has symmetry group F₄, while the other two are dual and have symmetry group H₄.

The Coxeter groups of type D_n, E₆, E₇, and E₈ are the symmetry groups of certain semiregular polytopes

Thorold Gosset

Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...

Affine Coxeter groups

The affine Coxeter groups form a second important series of Coxeter groups. These are not finite themselves, but each contains a normal

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

such that the corresponding quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

is finite. In each case, the quotient group is itself a Coxeter group, and the Coxeter graph is obtained from the Coxeter graph of the Coxeter group by adding an additional vertex and one or two additional edges. For example, for n ≥ 2, the graph consisting of n+1 vertices in a circle is obtained from A_n in this way, and the corresponding Coxeter group is the affine Weyl group of A_n. For n = 2, this can be pictured as the symmetry group of the standard tiling of the plane by equilateral triangles.

A list of the affine Coxeter groups follows:

Group symbol	Witt Ernst Witt Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine.... symbol	Bracket notation Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...	Related uniform tessellation(s)	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
	P_n+1	[3^[n+1]]	Simplectic honeycomb :Triangular tiling :Tetrahedral-octahedral honeycomb Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2....	...
	S_n+1	[4,3^n-2,3^1,1]	Demihypercubic honeycomb	...
	R_n+1	[4,3^n-1,4]	Hypercubic honeycomb	...
	Q_n+1	[ 3^1,1,3^n-3,3^1,1]	Demihypercubic honeycomb	...
	T₇	[3^2,2,2]	2₂₂
	T₈	[3^3,3,1]	3₃₁, 1₃₃
	T₉	[3^5,2,1]	5₂₁ 5 21 honeycomb In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure... , 2₅₁, 1₅₂
	U₅	[3,4,3,3]	16-cell honeycomb 24-cell honeycomb
	V₃	[6,3]	Hexagonal tiling and Triangular tiling
	W₂	[∞]	apeirogon Apeirogon An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...

The subscript is one less than the number of nodes in each case, since each of these groups was obtained by adding a node to a finite group's graph.

Hyperbolic Coxeter groups

There are infinitely many hyperbolic Coxeter groups describing reflection groups in hyperbolic space

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, notably including the hyperbolic triangle group

Triangle group

Partial orders

A choice of reflection generators gives rise to a length function

Length function

In mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.-Definition:...

l on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the word metric in the Cayley graph

Cayley graph

In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...

. An expression for v using l(v) generators is a reduced word. For example, the permutation (13) in S₃ has two reduced words, (12)(23)(12) and (23)(12)(23). The function

defines a map

generalizing the sign map for the symmetric group.

Using reduced words one may define three partial orders on the Coxeter group, the (left) weak order, the absolute order and the Bruhat order
Bruhat order
In mathematics, the Bruhat order is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.-History:The Bruhat order on the Schubert varieties of a flag manifold or Grassmannian...

(named for François Bruhat

François Bruhat

François Georges René Bruhat was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him....

). An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped. In the weak order, v ≥ u if some reduced word for v contains a reduced word for u as an initial segment. Indeed, the word length makes this into a graded poset

Graded poset

In mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to N compatible with the ordering such that whenever y covers x, then...

. The Hasse diagram

Hasse diagram

In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...

s corresponding to these orders are objects of study, and are related to the Cayley graph

Cayley graph

determined by the generators. The absolute order is defined analogously to the weak order, but with generating set/alphabet consisting of all conjugates of the Coxeter generators.

For example, the permutation (1 2 3) in S₃ has only one reduced word, (12)(23), so covers (12) and (23) in the Bruhat order but only covers (12) in the weak order.

Homology

Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group

Elementary Abelian group

In group theory, an elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime; in particular it is a p-group....

, i.e. it is isomorphic to the direct sum of several copies of the cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

Z₂. This may be restated in terms of the first homology group of W.

The Schur multiplier

Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...

M(W) (related to the second homology) was computed in for finite reflection groups and in for affine reflection groups, with a more unified account given in . In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family {W_n} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.