Parabolic subgroup of a reflection group

Summary

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 $\{1, 2, \ldots, n\}$, which is generated by the set $S = \{(1\ 2), (2\ 3), \ldots, (n - 1\ n)\}$ of adjacent transpositions, a subgroup is a standard parabolic subgroup if it is generated by a subset of $S$; equivalently, they are the groups of permutations that come from partitioning the set $\{1, 2, \ldots, n\}$ into parts $\{1, 2, \ldots, a_1\}$, $\{a_1 + 1, a_1 + 2, \ldots, a_1 + a_2\}$, etc., each consisting of a subset of one or more consecutive values, and permuting the entries of each set among itself. The parabolic subgroups of the symmetric group include the standard parabolic subgroups as well as all of their conjugates.