Misplaced Pages

This is an old revision of this page, as edited by JayBeeEll (talk | contribs) at 17:24, 8 January 2024 (→References). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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. 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 Coxeter & ???? harvtxt error: no target: CITEREFCoxeter???? (help), the finite real reflection groups (that is, those finite groups ......) are precisely the finite Coxeter groups. By extension of scalars, each such group is also a complex reflection group. In these groups, the two definitions above are not equivalent, as there are many more subspaces blahblahblah than subsets blahblahblah. This discepancy is harmonized as follows:

....

In dual Coxeter theory

Affine Coxeter groups

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

Footnotes

1. terminology: reflection vs. pseudoreflection, crg vs. unitary
2. Sometimes such subgroups are called isotropy groups.
3. Including the entire space V, as the empty intersection.

1. ^ Björner & Brenti (2005), §2.4.
2. ^ Humphreys (1990), §5.5.
3. Humphreys (1990), §1.10.
4. Humphreys (1990), §5.10.
5. Kane (2001), p. 60. sfnp error: no target: CITEREFKane2001 (help)
6. ^ Lehrer & Taylor (2009), p. 171.
7. Lehrer & Taylor (2009), §9.7.
8. Lehrer & Taylor (2009), §9.3.
9. ^ Broué (2010), §4.2.4.
10. ^ ????. sfnp error: no target: CITEREF???? (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
• 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