Parabolic subgroup of a reflection group

In the mathematical theory of reflection groups, the parabolic subgroups are a special kind of subgroup. In the symmetric group of permutations of the set ${\displaystyle \{1,2,\ldots ,n\}}$, which is generated by the set ${\displaystyle S=\{(1\ 2),(2\ 3),\ldots ,(n-1\ n)\}}$ of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of S.^[a] The parabolic subgroups of the symmetric group include the standard parabolic subgroups as well as all of their conjugates.

The symmetric group (generated by the adjacent transpositions) belongs to a larger family of reflection groups called Coxeter groups, each of which comes with a special generating set S. In this larger family, the standard parabolic subgroups are defined to be the subgroups generated by a subset of the special generating set. Separately, the symmetric group belongs to a larger family of reflection groups called complex reflection groups, which are defined in terms of their action on certain geometric spaces (finite dimensional complex vector spaces). In this family, a subgroup is parabolic if it consists of all elements of the group that fix a given subset of the space pointwise. In the case of groups that are both Coxeter groups and complex reflection groups, the parabolic subgroups (in the second sense) consist of the standard parabolic subgroups (in the first sense) and all of their conjugates.

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.

In addition to their role in geometry (where they arise as symmetry groups of regular polyhedra), reflection groups arise in the theory of algebraic groups, through their connection with Weyl groups. The parabolic subgroups are so-named because they correspond to parabolic subgroups of algebraic groups in this setting.
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