In the mathematical theory of reflection groups, a parabolic subgroup is 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, and the different definitions essentially coincide in the case of finite real reflection groups. For example, the parabolic subgroups of a reflection group have a natural indexing set and form a lattice when ordered by inclusion. Parabolic subgroups arise in the theory of algebraic groups, through their connection with Weyl groups.

In Coxeter groups

Suppose that W is a Coxeter group with simple reflections S.^{[lower-alpha 1]} For each subset I of S, let ${\displaystyle W_{I}}$ denote the subgroup of W generated by ${\displaystyle I}$. Such subgroups are called standard parabolic subgroups of W.^[2][3] In the extreme cases, ${\displaystyle W_{\varnothing }}$ is the trivial subgroup (containing just the identity element of W) and ${\displaystyle W_{S}=W}$.^[4]

The pair ${\displaystyle (W_{I},I)}$ is again a Coxeter group. Moreover, the Coxeter group structure on ${\displaystyle W_{I}}$ is compatible with that on W, in the following sense: if ${\displaystyle \ell _{S}}$ denotes the length function on W with respect to S (so that ${\displaystyle \ell _{S}(w)=k}$ if the element w of W can be written as a product of k elements of S and not fewer), then for every element w of ${\displaystyle W_{I}}$, one has that ${\displaystyle \ell _{S}(w)=\ell _{I}(w)}$. That is, the length of w is the same whether it is viewed as an element of W or of ${\displaystyle W_{I}}$.^[2][3] The same is true of the Bruhat order: if u and w are elements of ${\displaystyle W_{I}}$, then ${\displaystyle u\leq w}$ in the Bruhat order on ${\displaystyle W_{I}}$ if and only if ${\displaystyle u\leq w}$ in the Bruhat order on W.^[5]

If I and J are two subsets of S, then ${\displaystyle W_{I}=W_{J}}$ if and only if ${\displaystyle I=J}$, ${\displaystyle W_{I}\cap W_{J}=W_{I\cap J}}$, and the smallest group ${\displaystyle \langle W_{I},W_{J}\rangle }$ that contains both ${\displaystyle W_{I}}$ and ${\displaystyle W_{J}}$ is ${\displaystyle W_{I\cup J}}$. Consequently, the lattice of standard parabolic subgroups of W is a Boolean lattice.^[2][3]

Given a standard parabolic subgroup ${\displaystyle W_{I}}$ of a Coxeter group W, the cosets of ${\displaystyle W_{I}}$ in W have a particularly nice system of representatives: let ${\displaystyle W^{I}}$ denote the set

$${\displaystyle W^{I}=\{w\in W\colon \ell _{S}(ws)>\ell _{S}(w){\text{ for all }}s\in I\}}$$

of elements in W that do not have any element of I as a right descent.^{[lower-alpha 2]} Then for each ${\displaystyle w\in W}$, there are unique elements ${\displaystyle u\in W^{I}}$ and ${\displaystyle v\in W_{I}}$ such that ${\displaystyle w=uv}$. Moreover, this is a length-additive product, that is, ${\displaystyle \ell _{S}(w)=\ell _{S}(u)+\ell _{S}(v)}$. Furthermore, u is the element of minimum length in the coset ${\displaystyle wW_{I}}$.^[2][7] An analogous construction is valid for right cosets.^[8]

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.^[9] 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.^[10]

In complex reflection groups

Suppose that W is a complex reflection group^{[lower-alpha 3]} acting on a complex vector space V. For any subset ${\displaystyle A\subseteq V}$, let

$${\displaystyle W_{A}=\{w\in W\colon w(a)=a{\text{ for all }}a\in A\}}$$

be the subset of W consisting of those elements in W that fix each element of A.^{[lower-alpha 4]} Such a subgroup is called a parabolic subgroup of W.^[14] In the extreme cases, ${\displaystyle W_{\varnothing }=W_{\{0\}}=W}$ and ${\displaystyle W_{V}}$ is the trivial subgroup of W that contains only the identity element.

It follows from a theorem of Steinberg (1964) that each parabolic subgroup ${\displaystyle W_{A}}$ of a complex reflection group W is a reflection group, generated by the reflections in W that fix every point in A.^[15] Since W acts linearly on V, ${\displaystyle W_{A}=W_{\overline {A}}}$ where ${\displaystyle {\overline {A}}}$ is the span of A (that is, the smallest linear subspace of V that contains A).^[14] 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.^[16] The collection of all intersections of subsets of these hyperplanes,^{[lower-alpha 5]} partially ordered by inclusion, is a lattice ${\displaystyle L_{W}}$.^[17] The elements of the lattice are precisely the fixed spaces of the elements of W (that is, for each intersection I of reflecting hyperplanes, there is an element ${\displaystyle w\in W}$ such that ${\displaystyle \{v\in V\colon w(v)=v\}=I}$).^[18][19] The map that sends ${\displaystyle I\mapsto W_{I}}$ for ${\displaystyle I\in L_{W}}$ is an order-reversing bijection between subspaces in ${\displaystyle L_{W}}$ and parabolic subgroups of W.^[19]

Concordance of definitions in finite real reflection groups

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. By a theorem of H. S. M. Coxeter, W is a finite Coxeter groups.^{[lower-alpha 6][20]} By complexification, each such group is also a complex reflection group.^[21] 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.^[22]

Examples

The symmetric group ${\displaystyle S_{n}}$, which consists of all permutations of ${\displaystyle \{1,\ldots ,n\}}$, is a Coxeter group with respect to the set of adjacent transpositions ${\displaystyle (1,2)}$, ..., ${\displaystyle (n-1,n)}$. The standard parabolic subgroups of ${\displaystyle S_{n}}$ are the subgroups of the form ${\displaystyle S_{a_{1}}\times \cdots \times S_{a_{k}}}$, where ${\displaystyle a_{1},\ldots ,a_{k}}$ are positive integers of sum n, in which the first factor permutes the elements ${\displaystyle \{1,\ldots ,a_{1}\}}$ among themselves, the second factor permutes the elements ${\displaystyle \{a_{1}+1,\ldots ,a_{1}+a_{2}\}}$ among themselves, and so on.^[23][8]

The hyperoctahedral group ${\displaystyle S_{n}^{B}}$, which consists of all signed permutations of ${\displaystyle \{\pm 1,\ldots ,\pm n\}}$ (that is, the bijections w on that set such that ${\displaystyle w(-i)=-w(i)}$ for all i), has as its maximal standard parabolic subgroups the stabilizers of ${\displaystyle \{i+1,\ldots ,n\}}$ for ${\displaystyle i\in \{1,\ldots ,n\}}$.^[24]

In dual Coxeter theory

If W is a group and T is a subset of W, the pair ${\displaystyle (W,T)}$ is called a dual Coxeter system if there exists a subset S of T such that ${\displaystyle (W,S)}$ is a Coxeter system and

$${\displaystyle T=\{wsw^{-1}\colon w\in W,s\in S\},}$$

so that T is the set of all reflections (conjugates of the simple reflections) in W. For a dual Coxeter system ${\displaystyle (W,T)}$, a subgroup of W is said to be a parabolic subgroup if it is a standard parabolic (as in § In Coxeter groups) of ${\displaystyle (W,S)}$ for some choice of simple reflections S for ${\displaystyle (W,T)}$.^{[25][lower-alpha 7]}

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 ${\displaystyle (W,T)}$, but not all pairs of reflections are conjugate to each other.^[25] 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).^[27] The same result does not hold in general for infinite Coxeter groups.^[28]

Affine and crystallographic Coxeter groups

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.^[29] In fact, every standard parabolic subgroup of an affine Coxeter group is finite.^[30] 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

$${\displaystyle \{w\in W\colon w(a)=a{\text{ for all }}a\in A\}}$$

for some subset A of V.^[31]

If W is a crystallographic Coxeter group,^{[lower-alpha 8]} then every parabolic subgroup of W is also crystallographic.^[32]

Connection with the theory of algebraic groups

If G is an algebraic group and B is a Borel subgroup for G, then a parabolic subgroup of G is any subgroup that contains B. If furthermore G has a (B, N) pair, then the associated quotient group ${\displaystyle W=B/(B\cap N)}$ is a Coxeter group, called the Weyl group of G. Then the group G has a Bruhat decomposition ${\displaystyle G=\bigsqcup _{w\in W}BwB}$ into double cosets (where ${\displaystyle \sqcup }$ is the disjoint union), and the parabolic subgroups of G containing B are precisely the subgroups of the form

$${\displaystyle P_{J}=BW_{J}B}$$

where ${\displaystyle W_{J}}$ is a standard parabolic subgroup of W.^[33]

Footnotes

References

• Baumeister, Barbara; Dyer, Matthew; Stump, Christian; Wegener, Patrick (2014), "A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements", Proc. Amer. Math. Soc., B, 1: 149–154
• Baumeister, Barbara; Gobet, Thomas; Roberts, Kieran; Wegener, Patrick (2017), "On the Hurwitz action in finite Coxeter groups", J. Group Theory, 20 (1): 103–131
• Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, Springer, doi:10.1007/3-540-27596-7, ISBN 978-3540-442387, S2CID 115235335
• Broué, Michel (2010), Introduction to complex reflection groups and their braid groups, Lecture Notes in Mathematics, vol. 1988, Springer-Verlag, doi:10.1007/978-3-642-11175-4, ISBN 978-3-642-11174-7
• Digne, François; Michel, Jean (1991), Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, ISBN 0-521-40117-8
• Gobet, Thomas (2017), "On cycle decompositions in Coxeter groups", Sém. Lothar. Combin., 78B: Art. 45
• Humphreys, James E. (1990), Reflection groups and Coxeter groups, Cambridge University Press, doi:10.1017/CBO9780511623646, ISBN 0-521-37510-X, S2CID 121077209
• Kane, Richard (2001), Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, doi:10.1007/978-1-4757-3542-0, ISBN 0-387-98979-X, S2CID 119694827
• Lehrer, Gustav I.; Taylor, Donald E. (2009), Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20, Cambridge University Press, ISBN 978-0-521-74989-3
• Orlik, Peter; Terao, Hiroaki (1992), Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften, Springer, doi:10.1007/978-3-662-02772-1, ISBN 978-3-540-55259-8
• Reiner, Victor; Ripoll, Vivien; Stump, Christian (2017), "On non-conjugate Coxeter elements in well-generated reflection groups", Math. Z., 285 (3–4): 1041–1062
• Steinberg, Robert (1964), "Differential equations invariant under finite reflection groups", Transactions of the American Mathematical Society, 112: 392–400, doi:10.1090/S0002-9947-1964-0167535-3