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I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one infinite class and 34 exceptional cases. I have shown the result for the groups $G(r,1,n)$. To finish checking my conjecture, I need to know the "roots" associated to each of the exceptional groups (well, I guess just the Shephard groups).

Let me elaborate: Each complex reflection group is generated by generalized reflections, which are elements of $GL(V)$ with finite order that fix a hyperplane point-wise. I need to know what these hyperplanes are.

I'm looking for a list along the lines of what occurs in the "Plates" of Bourbaki's book on Lie Groups, or in Section 2.10 of Humphrey's book on real reflection groups.

Thanks.

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    $\begingroup$ Though I haven't gone far enough into this to know what's really important, I'd suggest you look at the softcover 2009 Cambridge Univ. Press. book by G.I. Lehrer and D.E. Taylor, Unitary Reflection Groups. They introduce "root systems" in appropriate generality at the end of Chapter 1, and later they develop detailed information based on the classification. What I don't know is whether you will easily locate the data you need, though they have a lot of tables at the end of the book. $\endgroup$ Aug 17, 2016 at 22:10
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    $\begingroup$ There is an ongoing project of Broué-Corran-Michel on roots for complex reflection groups. There is apparently no preprint yet but here are notes from a talk of Broué on the topic webusers.imj-prg.fr/~michel.broue/Serge2015.pdf I have not idea if this helps or not for your problem, but it might be worth having a look and maybe write an email to Jean Michel. $\endgroup$ Sep 7, 2016 at 12:54
  • $\begingroup$ In "Congruence classes of presentations for the complex reflection groups $G(m, p, n)$", Paragraph 1.4, Shi provides explicit representatives for the root system of $G(m,p,n)$. $\endgroup$ Nov 2, 2016 at 9:52

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