Parabolic subgroup of a reflection group

In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set, which is generated by the set of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of ; equivalently, they are the groups of permutations that come from partitioning the set into parts,, etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself. The parabolic subgroups of the symmetric group include the standard parabolic subgroups as well as all of their conjugates.
The symmetric group belongs to a larger family of reflection groups called Coxeter groups, each of which comes with a special generating set . In this larger family, a subgroup is a standard parabolic subgroup if it is generated by a subset of the special generating set. Separately, the symmetric group belongs to a larger family of reflection groups called complex reflection groups, which are defined in terms of their action on certain geometric spaces. In this family, a subgroup is parabolic if it consists of all elements of the group that fix a given subset of the space pointwise. In the case of groups that are both Coxeter groups and complex reflection groups, the parabolic subgroups consist of the standard parabolic subgroups and all of their conjugates.
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.
In addition to their role in geometry, reflection groups arise in the theory of algebraic groups, through their connection with Weyl groups. The parabolic subgroups are so-named because they correspond to parabolic subgroups of algebraic groups in this setting.

Background: reflection groups

In a Euclidean space, a reflection is a symmetry of the space across a mirror that fixes the vectors that lie on the mirror and sends the vectors orthogonal to the mirror to their negatives. A finite real reflection group is a finite group generated by reflections. For example, the symmetries of a regular polygon in the plane form a reflection 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 generalization.
Each finite real reflection group has the structure of a Coxeter group: this means that contains a subset of reflections such that generates, subject to relations of the form
where denotes the identity in and the are numbers that satisfy for and for. 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.

Definitions

In Coxeter groups

Suppose that is a Coxeter group with a finite set of simple reflections. For each subset of, let denote the subgroup of generated by. Such subgroups are called standard parabolic subgroups of the Coxeter system. In the extreme cases, is the trivial subgroup and
The pair is again a Coxeter system. Moreover, the Coxeter group structure on is compatible with that on, in the following sense: if denotes the length function on with respect to , then for every element of, one has that. That is, the length of is the same whether it is viewed as an element of or of. The same is true of the Bruhat order : if and are elements of, then in the Bruhat order on if and only if in the Bruhat order on.
If and are two subsets of, then if and only if. Also,, and the smallest group that contains both and is. Consequently, when the standard parabolic subgroups of are ordered by inclusion, the resulting lattice is the same as the Boolean lattice of all subsets of ordered by inclusion.
Given a standard parabolic subgroup of a Coxeter system, the cosets of in have a particularly nice system of representatives: let denote the set
of elements in that do not have any element of as a right descent. Then for each, there are unique elements and such that. Moreover, this is a length-additive product, that is,. Furthermore, is the element of minimum length in the coset. 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.
Every Coxeter system may be encoded as a Coxeter–Dynkin diagram, a graph whose nodes correspond to the elements of and whose edges encode the orders of the relations between pairs of generators. The Coxeter–Dynkin diagram of a standard parabolic subgroup arises 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, let
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, and is the trivial subgroup of that contains only the identity element.
It follows from a theorem of that each parabolic subgroup of a complex reflection group is a reflection group, generated by the reflections in that fix every point in. Since acts linearly on, where is the span of . In fact, there is a simple choice of subspaces that index the parabolic subgroups: each reflection in fixes a hyperplane pointwise, and the collection of all these hyperplanes is the reflection arrangement of. The collection of all possible intersections among the hyperplanes in the reflection arrangement, partially ordered by inclusion, is a lattice. The elements of the lattice are precisely the fixed spaces of the elements of . The map that sends for is an order-reversing bijection between subspaces in and parabolic subgroups of.

Compatibility of the 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, may be viewed as both a Coxeter group and as a complex reflection group in a natural way. For a real reflection group, the parabolic subgroups of are not all standard parabolic subgroups of , 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

The symmetric group, which consists of all permutations of, is a Coxeter group with respect to the set of adjacent transpositions,,...,. The standard parabolic subgroups of are the subgroups of the form, where are positive integers with sum, in which the first factor in the direct product permutes the elements among themselves, the second factor permutes the elements among themselves, and so on.
The hyperoctahedral group, which consists of all signed permutations of , is a Coxeter group with respect to the generating set. Its maximal standard parabolic subgroups are the setwise stabilizers of for.
In, the maximal standard parabolic subgroups are and . Under conjugation, these yield the additional parabolic subgroups and. The only other parabolic subgroups are the whole group and the trivial subgroup; when ordered by containment, these form a non-Boolean lattice, as depicted in the figure at right.

More general definitions in Coxeter theory

In a Coxeter system with a finite generating 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 is called a dual Coxeter system if there exists a subset of such that is a Coxeter system and
so that is the set of all reflections in. For a dual Coxeter system, a subgroup of is said to be a parabolic subgroup if it is a standard parabolic subgroup of 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 coincide with the parabolic subgroups with respect to any other simple system. However, even in finite examples, it may not be the case that all sets of simple reflections are conjugate: 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, but not all pairs of reflections are conjugate to each other. Nevertheless, if is finite, then the parabolic subgroups coincide with the parabolic subgroups in the classical sense. The same result does not hold in general for infinite Coxeter groups.