Irreducible Representations of Finite Coxeter Groups

Question

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.

Answer 1

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