# Computing Quantum Dimensions

Hi,
in "Jaegerâ€™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

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I'd recommend first learning the Weyl dimension formula for non-quantum groups (say by reading Fulton+Harris). Anyway, Scott Morrison's quantum groups package (see katlas.org/wiki/QuantumGroups%60) can compute this for any particular rep that you want. – Noah Snyder Dec 11 '10 at 19:56
THX! I recently installed Mathematica 7 so this should be no great problem. (But still, mind to email me some sample function call, say, G2 and the rep for the G2 invariant, to my email address fc3a501(at)uni-hamburg.de? I'm still a beginner at Mathematica and if I have to decompile the package for finding that call WeylDimension[group,rep] - or whatever it's called...) Question answered. Hauke – Hauke Reddmann Dec 13 '10 at 19:11