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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 ${\displaystyle S}$ is the set of simple reflections of the Coxeter group ${\displaystyle W}$. For each subset ${\displaystyle I}$ of ${\displaystyle S}$, let ${\displaystyle W_{I}}$ denote the subgroup of ${\displaystyle W}$ generated by ${\displaystyle I}$. Such subgroups are called standard parabolic subgroups of ${\displaystyle W}$.

In complex reflection groups

Concordance of definitions in finite real reflection groups

In dual Coxeter theory

Connections with Lie theory

References