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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. For example, the parabolic subgroups of a reflection group form a lattice when ordered by inclusion, they have a natural indexing set, and a parabolic subgroup of a parabolic subgroup is parabolic with respect to the whole group.

In Coxeter groups

Suppose that W is a Coxeter group with simple reflections S. 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. In the extreme cases, ${\displaystyle W_{\varnothing }}$ is the trivial subgroup (containing just the identity element of W) and ${\displaystyle W_{S}=S}$.

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 ${\displaystyle W_{I}}$. 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.

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. Moreover, if G is a standard parabolic subgroup of W and H is a standard parabolic subgroup of G, then H is a standard parabolic subgroup of W.

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: ....

In complex reflection groups

Suppose that W is a complex reflection group 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. Such a subgroup is called a parabolic subgroup of W. 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. 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). 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. The collection of all intersections of subsets of these hyperplanes, partially ordered by inclusion, is a lattice ${\displaystyle L_{W}}$. 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 ${\displaystyle \{v\in V\colon w(v)=v\}=I}$). 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.

Concordance of definitions in finite real reflection groups

By a theorem of H. S. M. Coxeter, the finite real reflection groups (that is, those finite groups of linear transformations on a finite-dimensional real Euclidean space that are generated by reflections) are precisely the finite Coxeter groups. By extension of scalars, each such group is also a complex reflection group. For a real reflection group W, the parabolic subgroups of W (viewed as a complex reflection group) are not all parabolic subgroups of W (when viewed as a Coxeter group, after specifying a Coxeter generating set S), as there are many more subspaces in the intersection lattice of its reflection arrangement than subsets of S. This discepancy is harmonized as follows:

....

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 (when W is viewed as a Coxeter group with simple reflections S). 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 {{Section link}}: required section parameter(s) missing) of ${\displaystyle (W,S)}$ for some choice of simple reflections S for ${\displaystyle (W,T)}$.

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 simple 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. 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).

Affine Coxeter groups

Connection with Lie theory and origin of the name "parabolic"

Footnotes

1. That is, W has a presentation of the form ${\displaystyle W=\langle S\mid (ss')^{m_{s,s'}}=1\rangle }$ where 1 denotes the identity in W and the ${\displaystyle m_{s,s'}}$ are numbers that satisfy ${\displaystyle m_{s,s}=1}$ for ${\displaystyle s\in S}$ (so each element of S is an involution) and ${\displaystyle m_{s,s'}\in \{2,3,\ldots ,\}\cup \{\infty \}}$ for ${\displaystyle s\neq s'\in S}$.
2. Such groups are also known as unitary reflection groups or complex pseudo-reflection groups in some sources. Similarly, sometimes complex reflections (linear transformations that fix a hyperplane pointwise) are called pseudo-reflections.
3. Sometimes such subgroups are called isotropy groups.
4. Including the entire space V, as the empty intersection.

1. Kane (2001), 6.1. sfnp error: no target: CITEREFKane2001 (help)
2. ^ Björner & Brenti (2005), §2.4.
3. ^ Humphreys (1990), §5.5.
4. Humphreys (1990), §1.10.
5. Humphreys (1990), §5.10.
6. Humphreys (1990), p. 66.
7. Kane (2001), p. 60. sfnp error: no target: CITEREFKane2001 (help)
8. ^ Lehrer & Taylor (2009), p. 171.
9. Lehrer & Taylor (2009), §9.7.
10. Orlik & Terao (1992), p. 215.
11. Orlik & Terao (1992), §2.1.
12. Lehrer & Taylor (2009), §9.3.
13. ^ Broué (2010), §4.2.4.
14. Kane (2001), p. 82. sfnp error: no target: CITEREFKane2001 (help)
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16. ^ Baumeister et al. (2014). sfnp error: no target: CITEREFBaumeisterDyerStumpWegener2014 (help)
17. Baumeister et al. (2017), Prop. 1.4, Cor. 4.4. sfnp error: no target: CITEREFBaumeisterGobetRobertsWegener2017 (help)

References

• 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
• Humphreys, James E. (1990), Reflection groups and Coxeter groups, Cambridge University Press, doi:10.1017/CBO9780511623646, ISBN 0-521-37510-X, S2CID 121077209
• 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
• 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

• Coxeter groups
• Reflection groups