I am trying to understand finite reflection groups. Given a connected finite reflection group generated by $m$ reflections and let $S$ be a set of simple roots. Let $I \subset S$ be a subset of the simple roots. What is the type of the subgroup generated by I?
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What do you mean by "type"? There isn't nearly enough information to determine the Lie type. – Matt Samuel Jun 22 '15 at 18:15
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In the case of $A_{n}$ the group seems to aways be $A_{|I|}$? – Jeff Jun 24 '15 at 15:54
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Not always. Always a direct product of type $A$ groups though. – Matt Samuel Jun 24 '15 at 15:55
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It's easy to tell what the type of such a subgroup will be from the Coxeter graph. These are called "standard parabolic subgroups" and their Coxeter graphs are just the induced subgraph corresponding to the given simple roots. Disjoint connected components commute, giving a direct product. – Matt Samuel Jun 24 '15 at 15:58