Parabolic subgroup of a reflection group: Difference between revisions

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	== In Coxeter groups ==		== In Coxeter groups ==

			Suppose that <math>(W, S)</math> is a ], that is, that <math>S</math> is the set of simple reflections of the ] <math>W</math>. For each subset <math>I</math> of <math>S</math>, let <math>W_I</math> denote the subgroup of <math>W</math> generated by <math>I</math>. Such subgroups are called ''standard parabolic subgroups'' of <math>W</math>.

	== In complex reflection groups ==		== In complex reflection groups ==

Revision as of 00:34, 7 January 2024

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 $(W,S)$ 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 $W_{I}$ denote the subgroup of $W$ generated by $I$ . Such subgroups are called standard parabolic subgroups of $W$ .

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