Parabolic subgroup of a reflection group

Template:Short description Template:Good article 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 $\{1,2,\dots ,n\}$, which is generated by the set $S=\{(12),(23),\dots ,(n-1n)\}$ of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of Template:Mvar; equivalently, they are the groups of permutations that come from partitioning the set $\{1,2,\dots ,n\}$ into parts $\{1,2,\dots ,a_1\}$, $\{a_1+1,a_1+2,\dots ,a_1+a_2\}$, etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself.Template:Efn The parabolic subgroups of the symmetric group include the standard parabolic subgroups as well as all of their conjugates.Template:Efn

The symmetric group belongs to a larger family of reflection groups called Coxeter groups, each of which comes with a special generating set Template:Mvar (generalizing the adjacent transpositions). In this larger family, a subgroup is a standard parabolic subgroup if it is generated by a subset of the special generating set Template:Mvar. 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 (finite dimensional complex vector 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 (in the second sense) consist of the standard parabolic subgroups (in the first sense) 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 (where they arise as symmetry groups of regular polyhedra), 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

The dihedral group Template:Math of symmetries of a square is a reflection group, generated by two reflections Template:Mvar (across the line Template:Math) and Template:Mvar (across the line Template:Math), subject to the relation that Template:Math is the identity.

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 whose dimension is one less than the dimension of the whole space) 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 Template:Mvar is a finite group generated by reflections (that is, every linear transformation in Template:Mvar is a composition of some of the reflections in Template:Mvar).Template:Sfnp 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.Template:Sfnp Finite real reflection groups can be generalized in various ways,Template:Sfnp and the definition of parabolic subgroup depends on the choice of generalization.

Each finite real reflection group Template:Mvar has the structure of a Coxeter group:Template:Sfnp this means that Template:Mvar contains a subset Template:Mvar of reflections (called simple reflections) such that Template:Mvar generates Template:Mvar, subject to relations of the form $$W=⟨S\mid (s{s}^{\prime }{)}^{m_{s,s^{\prime }}}=1⟩,$$ where Template:Math denotes the identity in Template:Mvar and the $m_{s,s^{\prime }}$ are numbers that satisfy $m_{s,s}=1$ for $s\in S$ and $m_{s,s^{\prime }}\in \{2,3,\dots \}$ for $s\ne s^{\prime }\in S$.Template:EfnTemplate:Sfnp 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.Template:Sfnp 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.Template:Efn 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.Template:SfnpTemplate:Sfnp

Definitions

In Coxeter groups

The lattice of standard parabolic subgroups of the symmetric group Template:Math, generated as a Coxeter group by the simple reflections Template:Math, Template:Math, and Template:Math (the adjacent transpositions), with identity element Template:Mvar

Suppose that Template:Mvar is a Coxeter group with a finite set Template:Mvar of simple reflections. For each subset Template:Mvar of Template:Mvar, let $W_I$ denote the subgroup of Template:Mvar generated by Template:Mvar. Such subgroups are called standard parabolic subgroups of the Coxeter system $(W,S)$.Template:SfnpTemplate:Sfnp In the extreme cases, $W_{\varnothing }$ is the trivial subgroup (containing just the identity element of Template:Mvar) and Template:Nowrap

The pair $(W_I,I)$ is again a Coxeter system. Moreover, the Coxeter group structure on $W_I$ is compatible with that on Template:Mvar, in the following sense: if ${\ell }_S$ denotes the length function on Template:Mvar with respect to Template:Mvar (so that ${\ell }_S(w)=k$ if the element Template:Mvar of Template:Mvar can be written as a product of Template:Mvar elements of Template:Mvar and not fewer), then for every element Template:Mvar of $W_I$, one has that ${\ell }_S(w)={\ell }_I(w)$. That is, the length of Template:Mvar is the same whether it is viewed as an element of Template:Mvar or of $W_I$.Template:SfnpTemplate:Sfnp The same is true of the Bruhat order (an important partial order on the elements of a Coxeter group): if Template:Mvar and Template:Mvar are elements of $W_I$, then $u\le w$ in the Bruhat order on $W_I$ if and only if $u\le w$ in the Bruhat order on Template:Mvar.Template:Sfnp

If Template:Mvar and Template:Mvar are two subsets of Template:Mvar, then $W_I=W_J$ if and only if $I=J$. Also, $W_I\cap W_J=W_{I\cap J}$, and the smallest group $⟨W_I,W_J⟩$ that contains both $W_I$ and $W_J$ is $W_{I\cup J}$. Consequently, when the standard parabolic subgroups of Template:Mvar are ordered by inclusion, the resulting lattice is the same as the Boolean lattice of all subsets of Template:Mvar ordered by inclusion.Template:SfnpTemplate:Sfnp

Given a standard parabolic subgroup $W_I$ of a Coxeter system $(W,S)$, the cosets of $W_I$ in Template:Mvar have a particularly nice system of representatives: let $W^I$ denote the set $$W^I=\{w\in W:{\ell }_S(ws)>{\ell }_S(w)\text{ for all }s\in I\}$$ of elements in Template:Mvar that do not have any element of Template:Mvar as a right descent.Template:Efn 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, Template:Mvar is the element of minimum length in the coset $w{W}_I$.Template:SfnpTemplate:Sfnp An analogous construction is valid for right cosets.Template:Sfnp The collection of all left cosets of standard parabolic subgroups is one possible construction of the Coxeter complex.Template:Sfnp

Every Coxeter system may be encoded as a Coxeter–Dynkin diagram, a graph whose nodes correspond to the elements of Template:Mvar and whose edges encode the orders $m_{s,s^{\prime }}$ 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.Template:Sfnp The only normal parabolic subgroups arise by taking a union of connected components of the diagram, and the whole group Template:Mvar is the direct product of the irreducible Coxeter groups that correspond to the components.Template:Sfnp

In complex reflection groups

The lattice of parabolic subgroups of the dihedral group Template:Math, 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.

Suppose that Template:Mvar is a complex reflection group acting on a complex vector space Template:Mvar. For any subset $A\subseteq V$, let $$W_A=\{w\in W:w(a)=a\text{ for all }a\in A\}$$ be the subset of Template:Mvar consisting of those elements in Template:Mvar that fix each element of Template:Mvar.Template:Efn Such a subgroup is called a parabolic subgroup of Template:Mvar.Template:Sfnp In the extreme cases, $W_{\varnothing }=W_{\{0\}}=W$ and $W_V$ is the trivial subgroup of Template:Mvar that contains only the identity element.

It follows from a theorem of Template:Harvtxt that each parabolic subgroup $W_A$ of a complex reflection group Template:Mvar is a reflection group, generated by the reflections in Template:Mvar that fix every point in Template:Mvar.Template:Sfnp Since Template:Mvar acts linearly on Template:Mvar, $W_A=W_{\bar{A}}$ where $\bar{A}$ is the span of Template:Mvar (that is, the smallest linear subspace of Template:Mvar that contains Template:Mvar).Template:Sfnp In fact, there is a simple choice of subspaces Template:Mvar that index the parabolic subgroups: each reflection in Template:Mvar fixes a hyperplane (that is, a subspace of Template:Mvar whose dimension is Template:Math less than that of Template:Mvar) pointwise, and the collection of all these hyperplanes is the reflection arrangement of Template:Mvar.Template:Sfnp The collection of all possible intersections among the hyperplanes in the reflection arrangement,Template:Efn partially ordered by inclusion, is a lattice $L_W$.Template:Sfnp The elements of the lattice are precisely the fixed spaces of the elements of Template:Mvar (that is, for each intersection Template:Mvar of reflecting hyperplanes, there is an element $w\in W$ such that $\{v\in V:w(v)=v\}=I$).Template:SfnpTemplate:Sfnp 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 Template:Mvar.Template:Sfnp

Compatibility of the definitions in finite real reflection groups

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

Examples

The lattice of parabolic subgroups of the group Template:Math, represented as signed permutations of Template:Math with identity Template:Mvar

The symmetric group $S_n$, which consists of all permutations of $\{1,\dots ,n\}$, is a Coxeter group with respect to the set of adjacent transpositions $(12)$, $(23)$, ..., $(n-1n)$. The standard parabolic subgroups of $S_n$ (which are also known as Young subgroups) are the subgroups of the form $S_{a_1}\times \cdots \times S_{a_k}$, where $a_1,\dots ,a_k$ are positive integers with sum Template:Mvar, in which the first factor in the direct product permutes the elements $\{1,\dots ,a_1\}$ among themselves, the second factor permutes the elements $\{a_1+1,\dots ,a_1+a_2\}$ among themselves, and so on.Template:SfnpTemplate:Sfnp

The hyperoctahedral group $S_n^B$, which consists of all signed permutations of $\{\pm 1,\dots ,\pm n\}$ (that is, the bijections Template:Mvar on that set such that $w(-i)=-w(i)$ for all Template:Mvar), is a Coxeter group with respect to the generating set $\{(1-1),(12)(-1-2),\dots ,(n-1n)(-n+1-n)\}$. Its maximal standard parabolic subgroups are the setwise stabilizers of $\{i,\dots ,n\}$ for $i\in \{1,\dots ,n\}$.Template:Sfnp

In $S_2^B$, the maximal standard parabolic subgroups are $\{\iota ,(1-1)\}$ (stabilizing the set Template:Tmath) and $\{\iota ,(12)(-1-2)\}$ (stabilizing the set Template:Tmath). Under conjugation, these yield the additional parabolic subgroups $\{\iota ,(2-2)\}$ and $\{\iota ,(1-2)(-12)\}$. 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. (The hyperoctahedral group $S_2^B$ is isomorphic to the dihedral group $D_{2\times 4}$ as Coxeter systems, which is why we get the same lattice of parabolic subgroups of the dihedral group $D_{2\times 4}$ in the previous figure.)

More general definitions in Coxeter theory

In a Coxeter system with a finite generating set Template:Mvar 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.Template:Sfnp

If Template:Mvar is a group and Template:Mvar is a subset of Template:Mvar, the pair $(W,T)$ is called a dual Coxeter system if there exists a subset Template:Mvar of Template:Mvar such that $(W,S)$ is a Coxeter system and $$T=\{ws{w}^{-1}:w\in W,s\in S\},$$ so that Template:Mvar is the set of all reflections (conjugates of the simple reflections) in Template:Mvar. For a dual Coxeter system $(W,T)$, a subgroup of Template:Mvar is said to be a parabolic subgroup if it is a standard parabolic subgroup (as in Template:Section link) of $(W,S)$ for some choice of simple reflections Template:Mvar for Template:Nowrap

The reflection arrangement of the dihedral group Template:Math. One chamber is shaded gray. Any two reflections form a Coxeter generating set, but the pair Template:Math is not conjugate to the pair Template:Math.

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, it may not be the case that all sets of simple reflections are conjugate: for example, if Template:Mvar is the dihedral group with Template:Math elements, viewed as symmetries of a regular pentagon, and Template:Mvar is the set of reflection symmetries of the polygon, then any pair of reflections in Template:Mvar forms a simple system for $(W,T)$, but not all pairs of reflections are conjugate to each other.Template:EfnTemplate:SfnpTemplate:Sfnp Nevertheless, if Template:Mvar 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 Template:Mvar).Template:Sfnp The same result does not hold in general for infinite Coxeter groups.Template:Sfnp

Affine and crystallographic Coxeter groups

Among the finite real reflection groups, some (called Weyl groups) have the property that there is a periodic arrangement of points in space (a lattice) that is preserved by the action of the group. (For example, the usual square lattice in the plane is preserved by the action of the dihedral group $D_{2\times 4}$, so it is a Weyl group; but no lattice in the plane is preserved by the group of symmetries of a regular pentagon.)Template:Sfnp Each Weyl group has an associated affine Coxeter group, an infinite Coxeter group generated by the reflections in the finite group together with a single reflection across a translated copy of one of the original reflecting hyperplanes.Template:Sfnp When Template:Mvar 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 Template:Mvar. In particular, the length functions on the finite and affine groups coincide.Template:Sfnp In fact, every standard parabolic subgroup of an affine Coxeter group is finite.Template:Sfnp As in the case of finite real reflection groups, when we consider the action of an affine Coxeter group Template:Mvar on a Euclidean space Template:Mvar, the conjugates of the standard parabolic subgroups of Template:Mvar are precisely the subgroups of the form $$\{w\in W:w(a)=a\text{ for all }a\in A\}$$ for some subset Template:Mvar of Template:Mvar.Template:Sfnp

Every (possibly infinite) Coxeter group has a special representation called its geometric representation.Template:SfnpTemplate:Sfnp Generalizing the definition of Weyl group, we say that a (possibly infinite) Coxeter group is crystallographic if it stabilizes a lattice in its geometric representation. Then every parabolic subgroup of a crystallographic Coxeter group is also crystallographic.Template:Sfnp

Connection with the theory of algebraic and Lie groups

The terminology parabolic subgroup arises from a connection between reflection groups and algebraic groups, such as the general linear group of $n\times n$ invertible matrices over a field; in general, an algebraic variety on which there is a compatible group structure.Template:Sfnp Each algebraic group satisfying certain technical conditionsTemplate:Efn has a special class of subgroups called Borel subgroups; in the general linear group, one such subgroup is the subgroup of invertible lower-triangular matrices, while in general they are the Zariski closed and connected solvable algebraic subgroups.Template:Sfnp If Template:Mvar is an algebraic group and Template:Mvar is a Borel subgroup for Template:Mvar, then a standard parabolic subgroup of Template:Mvar relative to Template:Mvar is any subgroup that contains Template:Mvar, and a parabolic subgroup is any subgroup that is a standard parabolic relative to some Borel subgroup, or equivalently (since all Borel subgroups are conjugate) any subgroup conjugate to a standard parabolic relative to Template:Mvar.Template:Sfnp (This use of the phrase "parabolic subgroup" was introduced by Roger Godement in his paper Template:Harvtxt.Template:Sfnp) These are connected to parabolic subgroups of reflection groups, as follows.

Inside Template:Mvar, let Template:Mvar be a maximal torus, that is, a maximal subgroup isomorphic to $k^n$, where Template:Mvar is the underlying field.Template:Sfnp For example, in the case of the general linear group with Template:Mvar the lower-triangular matrices, one can take Template:Mvar to be the subgroup of invertible diagonal matrices.Template:Sfnp Let Template:Mvar be the normalizer of Template:Mvar in Template:Mvar; in the case of the general linear group, this is the subgroup of monomial matrices (matrices with exactly one nonzero entry in every row and column).Template:Sfnp Then $(B,N)$ is a [[(B, N) pair|Template:Math pair]] for Template:Mvar, and the associated quotient group $W=N/T$ is a Coxeter group, called the Weyl group of Template:Mvar.Template:SfnpTemplate:Sfnp In the case of the general linear group, Template:Mvar is the symmetric group $S_n$.Template:Sfnp The group Template:Mvar has a Bruhat decomposition $G=\underset{w\in W}{⨆}BwB$, where $\bigsqcup$ is the disjoint union and $BwB=\{b_{1}w{b}_2\mid b_1,b_2\in B\}$ is a double coset of Template:Mvar, and the standard parabolic subgroups of Template:Mvar with respect to Template:Mvar are precisely the subgroups of the form $$P_J=B{W}_{J}B=\underset{w\in W_J}{⨆}BwB,$$ where $W_J$ is a standard parabolic subgroup of Template:Mvar.Template:Sfnp

For example, when $G={\mathrm{G}\mathrm{L}}_4(k)$ is the general linear group of Template:Math invertible matrices over the field Template:Mvar, the associated Weyl group is the symmetric group $W=S_4$ of Template:Math permutation matrices. Taking the Borel subgroup Template:Mvar of lower-triangular invertible matrices, the corresponding Coxeter generating set consists of $$s_1=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],s_2=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right],\text{ and }{s}_3=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right].$$ Then for each subset Template:Mvar of $S=\{s_1,s_2,s_3\}$, the corresponding parabolic subgroups of Template:Mvar are $$B=B{W}_{\varnothing }B=\left\{\left[\begin{array}{cccc}\ast & 0& 0& 0\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & \ast \end{array}\right]\right\},B{W}_{\{s_1\}}B=\left\{\left[\begin{array}{cccc}\ast & \ast & 0& 0\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & \ast \end{array}\right]\right\},$$$$B{W}_{\{s_2\}}B=\left\{\left[\begin{array}{cccc}\ast & 0& 0& 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & \ast \end{array}\right]\right\},B{W}_{\{s_3\}}B=\left\{\left[\begin{array}{cccc}\ast & 0& 0& 0\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \end{array}\right]\right\},$$$$B{W}_{\{s_1,s_2\}}B=\left\{\left[\begin{array}{cccc}\ast & \ast & \ast & 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & 0\\ \ast & \ast & \ast & \ast \end{array}\right]\right\},B{W}_{\{s_1,s_3\}}B=\left\{\left[\begin{array}{cccc}\ast & \ast & 0& 0\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \end{array}\right]\right\},$$$$B{W}_{\{s_2,s_3\}}B=\left\{\left[\begin{array}{cccc}\ast & 0& 0& 0\\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \end{array}\right]\right\},\text{ and }BWB=G,$$ where the starred entries can take any values in Template:Mvar (provided the resulting matrix is invertible).Template:Sfnp

A similar account could be given in terms of Lie groups (groups with a natural structure as differentiable manifolds rather than as algebraic varieties). In this context the Borel subgroups are also defined to be the maximal closed and connected solvable subgroups, and the parabolic subgroups are the subgroups that contain a Borel subgroup.Template:Sfnp In the case the general linear group over the real or complex numbers viewed as a Lie group, the standard parabolic subgroups are the same subgroups as when it is viewed as an algebraic group, namely, they are the subgroups of block triangular matrices.Template:Sfnp

Parabolic closures

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

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. (The name "generalized braid group" arises from the fact that, in the special case $W=S_n$ is the symmetric group, the associated Artin–Tits group is the braid group on Template:Mvar strands.)Template:Sfnp 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 Template:Mvar, and a parabolic subgroup is any subgroup conjugate to a standard parabolic.Template:Sfnp

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

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

References

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