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The Coxeter group is defined as

$$S = \langle s_i : s_i^2 = (s_i s_j)^{m_{ij}} = 1 \rangle $$

Does it have an irreducible representation of dimension >2 for $S$ finite?

Is there a reference on this subject i.e. to find irreps of finite $S$?

For the lowest two cases: $$ i=1; S=C_2$$ $$ i=1,2; S=Dih(2m)$$ so that there is no irrep of dim >2 .

Thanks.

Herband
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    Has the standard literature not proven useful? I assume you know that finite Coxeter groups are, for all intents and purposes, just finite reflection groups. There is a summary of some of the representation theory of finite Coxeter group in Humphrey's book "Reflection Groups and Coxeter Groups" for example. – Alex Youcis Mar 09 '13 at 21:33
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    Yes, I do have this book and I do also know that finite Coxeter groups correspond to Euclidean reflection groups. But I do not see them mentioned in the classification of finite simple groups and thus I asked. Would you please give me the pages as the book seems a bit vast for me at the moment and I did not want to conclude with one reading. – Herband Mar 09 '13 at 21:44
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    @Herbrand I don't have the book with me right now. That said this looks promising (http://mathoverflow.net/questions/31868/representations-of-finite-coxeter-groups) :) – Alex Youcis Mar 09 '13 at 21:46
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    @Alex Based on your comment and the reference I found the details on Chap 9 of the book and pp117-121. Thank you very much. – Herband Mar 09 '13 at 22:43
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    @Herbrand Should I post this as an answer, so that you can mark it solved? – Alex Youcis Mar 09 '13 at 22:45
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    @AlexYoucis, yes, please proceed to post. – Herband Mar 09 '13 at 22:59

1 Answers1

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The standard reference for such matters is Jim Humphrey's "Reflection Groups and Coxeter Groups". This link might also be of interest.

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