Parabolic subgroup of a reflection group

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.

In a Euclidean space (such as the Euclidean plane, ordinary three-dimensional space, or their 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.

[1] 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.

A separate generalization is to consider the geometric action on vector spaces whose underlying field is not the real numbers.

[1] 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.

[6][7] Suppose that W is a Coxeter group with a finite set S of simple reflections.

[14] The collection of all left cosets of standard parabolic subgroups is one possible construction of the Coxeter complex.

[15] 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.

[16] The only normal parabolic subgroups arise by taking a union of connected components of the diagram, and the whole group W is the direct product of the irreducible Coxeter groups that correspond to the components.

[19] In fact, there is a simple choice of subspaces A that index the parabolic subgroups: each reflection in W fixes a hyperplane (that is, a subspace of V whose dimension is 1 less than that of V) pointwise, and the collection of all these hyperplanes is the reflection arrangement of W.[21] The collection of all intersections of subsets of these hyperplanes,[e] partially ordered by inclusion, is a lattice

and parabolic subgroups of W.[24] Let W be a finite real reflection group; that is, W is a finite group of linear transformations on a finite-dimensional real Euclidean space that is generated by orthogonal reflections.

For a real reflection group W, the parabolic subgroups of W (viewed as a complex reflection group) are not all standard parabolic subgroups of W (when viewed as a Coxeter group, after specifying a fixed Coxeter generating set S), as there are many more subspaces in the intersection lattice of its reflection arrangement than subsets of S. However, in a finite real reflection group W, every parabolic subgroup is conjugate to a standard parabolic subgroup with respect to S.[25] The symmetric group

, is a Coxeter group with respect to the set of adjacent transpositions

are positive integers with sum n, in which the first factor in the direct product permutes the elements

[27] In a Coxeter group generated by a finite set S of simple reflections, one may define a parabolic subgroup to be any conjugate of a standard parabolic subgroup.

The same does not hold in general for Coxeter groups of infinite rank.

is called a dual Coxeter system if there exists a subset S of T such that

[29][f] 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 W is the dihedral group with 10 elements, viewed as symmetries of a regular pentagon, and T is the set of reflection symmetries of the polygon, then any pair of reflections in T forms a simple system for

[29] Nevertheless, if W 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 S).

[31] The same result does not hold in general for infinite Coxeter groups.

[32] When W 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 W. In particular, the length functions on the finite and affine groups coincide.

[33] In fact, every standard parabolic subgroup of an affine Coxeter group is finite.

[34] As in the case of finite real reflection groups, when we consider the action of an affine Coxeter group W on a Euclidean space V, the conjugates of the standard parabolic subgroups of W are precisely the subgroups of the form

[28] The same analysis applies to complex reflection groups, where the parabolic closure of X is also the pointwise stabiliser of the space of fixed points of X.

[39] The same does not hold for Coxeter groups of infinite rank.

Consequently, the parabolic subgroups of B form a lattice under inclusion.

[j] This definition naturally extends to finite complex reflection groups.

Eight subgroups of the symmetric group of permutations of the four-element set {1, 2, 3, 4}. Each subgroup is generated by some of the three adjacent transpositions (1 2), (2 3), (3 4). The subgroups are ordered by inclusion, with the trivial group (containing just the identity permutation) at the bottom, the entire symmetric group at the top, and the other six in-between; edges are drawn to connect smaller subgroups to the larger groups that contain them.
The lattice of standard parabolic subgroups of the symmetric group S 4 , generated as a Coxeter group by the simple reflections s 1 = (1 2) , s 2 = (2 3) , and s 3 = (3 4) (the adjacent transpositions ), with identity element ι
On the left, a square is drawn, along with its four lines of symmetry; the lines are labeled by their equations (x = y, y = 0, etc.). On the right, the subspaces fixed by the different symmetries are listed by reverse-inclusion, with the entire plane at the bottom, then the four symmetry lines above it, and at top the single point (0, 0).
The lattice of parabolic subgroups of the dihedral group D 2×4 , represented as a real reflection group, consists of the trivial subgroup, the four two-element subgroups generated by a single reflection, and the entire group. Ordered by inclusion, they give the same lattice as the lattice of fixed spaces ordered by reverse-inclusion.
The lattice of parabolic subgroups of the group S B
4 , represented as signed permutations of {−2, −1, 1, 2}, with identity ι