Parabolic subgroup of a reflection group explained

In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. The precise definition of which subgroups are parabolic depends on context—for example, whether one is discussing general Coxeter groups or complex reflection groups—but in all cases the collection of parabolic subgroups exhibits important good behaviors. For example, the parabolic subgroups of a reflection group have a natural indexing set and form a lattice when ordered by inclusion. The different definitions of parabolic subgroups essentially coincide in the case of finite real reflection groups. Parabolic subgroups arise in the theory of algebraic groups, through their connection with Weyl groups.

Background: reflection groups

In a Euclidean space (such as the Euclidean plane, ordinary three-dimensional space, or higher-dimensional analogues), a reflection is a symmetry of the space across a mirror (technically, across a subspace of dimension one smaller than the whole space) that fixes the vectors that lie on the mirror and send the vectors orthogonal to the mirror to their negatives. A finite real reflection group is a finite group generated by reflections (that is, every linear transformation in is a composition of some of the reflections in). For example, the symmetries of a regular polygon in the plane form a reflection group (called the dihedral group), because each rotation symmetry of the polygon is a composition of two reflections. Finite real reflection groups can be generalized in various ways, and the definition of parabolic subgroup depends on the choice of definition.

Each finite real reflection group has the structure of a Coxeter group: this means that contains a subset of reflections (called simple reflections) such that generates, subject to relations of the form $$W\; =\; \backslash langle\; S\; \backslash mid\; (s\; s\text{'})^\; =\; 1\; \backslash rangle,$$ where denotes the identity in and the

m _{s,} are numbers that satisfy m _{s,} = 1 for s \in S and m _{s,} \in \{ 2 , 3 , \ldots\} for s ≠ s' \in S . Thus, the Coxeter groups form one generalization of finite real reflection groups.

A separate generalization is to consider the geometric action on vector spaces whose underlying field is not the real numbers. Especially, if one replaces the real numbers with the complex numbers, with a corresponding generalization of the notion of a reflection, one arrives at the definition of a complex reflection group. Every real reflection group can be complexified to give a complex reflection group, so the complex reflection groups form another generalization of finite real reflection groups.

In Coxeter groups

Suppose that is a Coxeter group with a finite set of simple reflections. For each subset of, let

W _{I} denote the subgroup of generated by I . Such subgroups are called standard parabolic subgroups of . In the extreme cases, W _{\varnothing} is the trivial subgroup (containing just the identity element of) and

The pair

(W _{I,} I) is again a Coxeter group. Moreover, the Coxeter group structure on W _{I} is compatible with that on, in the following sense: if \ell _{S} denotes the length function on with respect to (so that \ell _{S(w)} = k if the element of can be written as a product of elements of and not fewer), then for every element of W _{I} , one has that \ell _{S(w)} = \ell _{I(w)} . That is, the length of is the same whether it is viewed as an element of or of W _{I} . The same is true of the Bruhat order: if and are elements of W _{I} , then u \leq w in the Bruhat order on W _{I} if and only if u \leq w in the Bruhat order on .

If and are two subsets of, then

W _{I} = W _{J} if and only if I = J , W _{I} \cap W _{J} = W _{I\cap} , and the smallest group \langle W _{I,} W _{J} \rangle that contains both W _{I} and W _{J} is W _{I} . Consequently, the lattice of standard parabolic subgroups of is a Boolean lattice.

Given a standard parabolic subgroup

W _{I} of a Coxeter group, the cosets of W _{I} in have a particularly nice system of representatives: let W ^{I} denote the set$$W^I\; =\; \backslash$$of elements in that do not have any element of as a right descent. Then for each w \in W , there are unique elements u \in W ^{I} and v \in W _{I} such that w = uv . Moreover, this is a length-additive product, that is, \ell _{S(w)} = \ell _{S(u)} + \ell _{S(v)} . Furthermore, is the element of minimum length in the coset w W _{I} . An analogous construction is valid for right cosets. The collection of all left cosets of standard parabolic subgroups is one possible construction of the Coxeter complex.

In terms of the Coxeter–Dynkin diagram, the standard parabolic subgroups arise by taking a subset of the nodes of the diagram and the edges induced between those nodes, erasing all others. The only normal parabolic subgroups arise by taking a union of connected components of the diagram, and the whole group is the direct product of the irreducible Coxeter groups that correspond to the components.

In complex reflection groups

Suppose that is a complex reflection group acting on a complex vector space . For any subset

A \subseteq V , let $$W\_A\; =\; \backslash$$be the subset of consisting of those elements in that fix each element of . Such a subgroup is called a parabolic subgroup of . In the extreme cases, W _{\varnothing} = W _{\{0\} } = W and W _{V} is the trivial subgroup of that contains only the identity element.

It follows from a theorem of that each parabolic subgroup

W _{A} of a complex reflection group is a reflection group, generated by the reflections in that fix every point in . Since acts linearly on, W _{A} = W _{\overline{A} } where \overline{A} is the span of (that is, the smallest linear subspace of that contains). In fact, there is a simple choice of subspaces that index the parabolic subgroups: each reflection in fixes a hyperplane (that is, a subspace of whose dimension is less than that of) pointwise, and the collection of all these hyperplanes is the reflection arrangement of . The collection of all intersections of subsets of these hyperplanes, partially ordered by inclusion, is a lattice L _{W} . The elements of the lattice are precisely the fixed spaces of the elements of (that is, for each intersection of reflecting hyperplanes, there is an element w \in W such that \{v \in V \colon w(v) = v\} = I ). The map that sends I \mapsto W _{I} for I \in L _{W} is an order-reversing bijection between subspaces in L _{W} and parabolic subgroups of .

Concordance of definitions in finite real reflection groups

Let be a finite real reflection group; that is, is a finite group of linear transformations on a finite-dimensional real Euclidean space that is generated by orthogonal reflections. As mentioned above (see), may be viewed as both a Coxeter group and as a complex reflection group. For a real reflection group, the parabolic subgroups of (viewed as a complex reflection group) are not all standard parabolic subgroups of (when viewed as a Coxeter group, after specifying a fixed Coxeter generating set), as there are many more subspaces in the intersection lattice of its reflection arrangement than subsets of . However, in a finite real reflection group, every parabolic subgroup is conjugate to a standard parabolic subgroup with respect to .

Examples

S _{n} , which consists of all permutations of \{ 1 , \ldots, n\} , is a Coxeter group with respect to the set of adjacent transpositions ( 1 2 ) , ..., (n - 1 n) . The standard parabolic subgroups of S _{n} (which are also known as Young subgroups) are the subgroups of the form S _{a _{1}} x … x S _{a _{k}} , where a _{1,} \ldots, a _{k} are positive integers with sum, in which the first factor in the direct product permutes the elements \{ 1 , \ldots, a _{1\}} among themselves, the second factor permutes the elements \{a _{1} + 1 , \ldots, a _{1} + a _{2\}} among themselves, and so on.S ^{B} _{n} , which consists of all signed permutations of \{\pm 1 , \ldots, \pm n\} (that is, the bijections on that set such that w( - i) = - w(i) for all), has as its maximal standard parabolic subgroups the stabilizers of \{i + 1 , \ldots, n\} for i \in \{ 1 , \ldots, n\} .

More general definitions in Coxeter theory

In a Coxeter group generated by a finite set of simple reflections, one may define a parabolic subgroup to be any conjugate of a standard parabolic subgroup. Under this definition, it is still true that the intersection of any two parabolic subgroups is a parabolic subgroup. The same does not hold in general for Coxeter groups of infinite rank.

If is a group and is a subset of, the pair

(W, T) is called a dual Coxeter system if there exists a subset of such that (W, S) is a Coxeter system and$$T\; =\; \backslash ,$$so that is the set of all reflections (conjugates of the simple reflections) in . For a dual Coxeter system (W, T) , a subgroup of is said to be a parabolic subgroup if it is a standard parabolic (as in) of (W, S) for some choice of simple reflections for

In some dual Coxeter systems, all sets of simple reflections are conjugate to each other; in this case, the parabolic subgroups with respect to one simple system (that is, the conjugates of the standard parabolic subgroups) coincide with the parabolic subgroups with respect to any other simple system. However, even in finite examples, this may not hold: for example, if is the dihedral group with elements, viewed as symmetries of a regular pentagon, and is the set of reflection symmetries of the polygon, then any pair of reflections in forms a simple system for

(W, T) , but not all pairs of reflections are conjugate to each other. Nevertheless, if is finite, then the parabolic subgroups (in the sense above) coincide with the parabolic subgroups in the classical sense (that is, the conjugates of the standard parabolic subgroups with respect to a single, fixed, choice of simple reflections). The same result does not hold in general for infinite Coxeter groups.

Affine and crystallographic Coxeter groups

When is an affine Coxeter group, the associated finite Weyl group is always a maximal parabolic subgroup, whose Coxeter–Dynkin diagram is the result of removing one node from the diagram of . In particular, the length functions on the finite and affine groups coincide. In fact, every standard parabolic subgroup of an affine Coxeter group is finite. As in the case of finite real reflection groups, when we consider the action of an affine Coxeter group on a Euclidean space, the conjugates of the standard parabolic subgroups of are precisely the subgroups of the form$$\backslash$$for some subset of .

If is a crystallographic Coxeter group, then every parabolic subgroup of is also crystallographic.

Connection with the theory of algebraic groups

If is an algebraic group and is a Borel subgroup for, then a parabolic subgroup of is any subgroup that contains . If furthermore has a pair, then the associated quotient group

W = B / (B \cap N) is a Coxeter group, called the Weyl group of . Then the group has a Bruhat decomposition G = s qcup _{w} BwB into double cosets (where \sqcup is the disjoint union), and the parabolic subgroups of containing are precisely the subgroups of the form$$P\_J\; =\; BW\_JB$$where W _{J} is a standard parabolic subgroup of .

Parabolic closures

Suppose is a Coxeter group of finite rank (that is, the set of simple generators is finite). Given any subset of, one may define the parabolic closure of to be the intersection of all parabolic subgroups containing . As mentioned above, in this case the intersection of any two parabolic subgroups of is again a parabolic subgroup of, and consequently the parabolic closure of is a parabolic subgroup of ; in particular, it is the (unique) minimal parabolic subgroup of containing . The same analysis applies to complex reflection groups, where the parabolic closure of is also the pointwise stabiliser of the space of fixed points of . The same does not hold for Coxeter groups of infinite rank.

Braid groups

Each Coxeter group is associated to another group called its Artin–Tits group or generalized braid group, which is defined by omitting the relations

s ^{2} = 1 for each generator s \in S from its Coxeter presentation. Although generalized braid groups are not reflection groups, they inherit a notion of parabolic subgroups: a standard parabolic subgroup of a generalized braid group is a subgroup generated by a subset of the standard generating set, and a parabolic subgroup is any subgroup conjugate to a standard parabolic.

A generalized braid group is said to be of spherical type if the associated Coxeter group is finite. If is a generalized braid group of spherical type, then the intersection of any two parabolic subgroups of is also a parabolic subgroup. Consequently, the parabolic subgroups of form a lattice under inclusion.

For a finite real reflection group, the associated generalized braid group may be defined in purely topological language, without referring to a particular group presentation. This definition naturally extends to finite complex reflection groups. Parabolic subgroups can also be defined in this setting.