Hi,
in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg
(arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the
"Discussion" part. Evidently quantum groups (q-deformed Hopf algebras
if I read correctly) have representations, and those representations have
quantum dimensions, and those can computed by some Weyl formula.
Yup, I can parrot the lingo :-) but I couldn't compute an actual quantum
dimension even at gunpoint.
Now there is a deep connection between quantum algebras and knot theory.
I'm an amateur dabbling in knot theory and alone knowing some
quantum dimensions (which "are", by the connection, the value of a knot
polynome for a single loop) for assorted groups and representations
would be invaluable for my project of classifying S matrices.
(Example: For the group G2 the result is q^5+q^4+q+1+1/q+1/q^4+1/q^5,
and armed with nothing more than this knowledge, I could construct
an Kauffman abstract tensor model for Kuperbergs G2 invariant. It's no
impressive result, any expert could do this using higher math, but I
did it just with some tensor equations even I can understand.)
Kuperbergs paper(s) was the only one I know where explicite values for quantum dimensions were given at all, thus my plea: Can somebody compute quantum dimensions for me? The classification of simple groups is complete, so if there are finitely many representations too (I don't know) it's a finite task :-) A reference suffices too...if my university has the book.
Hauke Reddmann
