Parabolic subgroup of a reflection group

Template:Short description 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 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. 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 definition.

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

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 $I$. Such subgroups are called standard parabolic subgroups of Template:Mvar.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 group. 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: 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 $I=J$, $W_I\cap W_J=W_{I\cap J}$, and the smallest group $⟨W_I,W_J⟩$ that contains both $W_I$ and $W_J$ is $W_{I\cup J}$. Consequently, the lattice of standard parabolic subgroups of Template:Mvar is a Boolean lattice.Template:SfnpTemplate:Sfnp

Given a standard parabolic subgroup $W_I$ of a Coxeter group Template:Mvar, 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

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.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 intersections of subsets of these hyperplanes,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

Concordance of 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. 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 $(12)$, ..., $(n-1n)$. 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), has as its maximal standard parabolic subgroups the stabilizers of $\{i+1,\dots ,n\}$ for $i\in \{1,\dots ,n\}$.Template:Sfnp

More general definitions in Coxeter theory

In a Coxeter group generated by a finite 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 (as in Template:Section link) of $(W,S)$ for some choice of simple reflections Template:Mvar for Template:Nowrap

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 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: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

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

If Template:Mvar is a crystallographic Coxeter group,Template:Efn then every parabolic subgroup of Template:Mvar is also crystallographic.Template:Sfnp

Connection with the theory of algebraic groups

If Template:Mvar is an algebraic group and Template:Mvar is a Borel subgroup for Template:Mvar, then a parabolic subgroup of Template:Mvar is any subgroup that contains Template:Mvar.Template:Efn If furthermore Template:Mvar has a [[(B, N) pair|Template:Math pair]], then the associated quotient group $W=B/(B\cap N)$ is a Coxeter group, called the Weyl group of Template:Mvar. Then the group Template:Mvar has a Bruhat decomposition $G=\underset{w\in W}{⨆}BwB$ into double cosets (where $\bigsqcup$ is the disjoint union), and the parabolic subgroups of Template:Mvar containing Template:Mvar are precisely the subgroups of the form $$P_J=B{W}_{J}B$$ where $W_J$ is a standard parabolic subgroup of Template:Mvar.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.Template:EfnTemplate: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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