Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman). I would like to know if one can generate E6-E8 Lie groups as Coxeter groups in this way (representing their elements as a product of reflexions in the mirrors of such a 3D Weyl group, associated to Platonic groups of symmetry, as a root system). I recall that there is an article studying the paths of multiple reflections in such Weyl systems as kaleidoscopes; does anybody know it?
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3$\begingroup$ No, Lie groups cannot be realized as Coxeter groups. They are very different things (though related in certain ways…) $\endgroup$– Sam HopkinsCommented Jan 23, 2023 at 22:31
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$\begingroup$ Is there a way to relate the groupoid of reflection paths in Weil chambers and the exceptional Lie group? or it rather has to do with its representation theory? $\endgroup$– Lucian IonescuCommented Jan 31, 2023 at 2:11
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