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In the mathematical theory of reflection groups, a parabolic subgroup is a special kind of well behaved 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 Coxeter groups

Suppose that ${\displaystyle (W,S)}$ is a Coxeter system, that is, that S is the set of simple reflections of the Coxeter group W. For each subset I of S, let ${\displaystyle W_{I}}$ denote the subgroup of W generated by ${\displaystyle I}$. Such subgroups are called 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 parabolic subgroups of W is a Boolean lattice.

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.

Concordance of definitions in finite real reflection groups

In dual Coxeter theory

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

Footnotes

1. ^ Björner & Brenti (2005), §2.4.
2. ^ Humphreys (1990), §5.5.
3. Humphreys (1990), §1.10.
4. Humphreys (1990), §5.10.
5. Lehrer & Taylor (2009), p. 171.
6. Lehrer & Taylor (2009), §9.7.

References

• Björner, Anders; Brenti, Francesco (2005), Combinatorics of Coxeter groups, Springer, doi:10.1007/3-540-27596-7, ISBN 978-3540-442387, S2CID 115235335
• 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
• 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