I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the construction fail (maybe there is not even an associated quantum group)?

BTW, you would already help me by filling out the gap:

$I_2(6)=G_2: 7\bigotimes7=1\bigoplus7\bigoplus14\bigoplus27$

$I_2(5): R\bigotimes R=...\bigoplus ...$ (also for the defining irrep R)

$I_2(4)=BC_2: 5\bigotimes5=1\bigoplus10\bigoplus14$

`$H_3, H_4$`

etc. Concerning the "defining irrep"Rfor the dihedral group of order 10, what is meant? For other types you are dealing with irreps of the corresponding Lie group, though`$BC_2$`

gives a smaller choice of dimension 4. – Jim Humphreys Nov 21 '11 at 14:34