# Pictorial explanation of Dynkin index and quadratic Casimir?

Draw a colored loop. You explained quantum dimension. :-) OK, now you may use trivalent nodes and knot crossings too (with edges colored by irreps). You can now pictorialize 6j symbols, writhe normalizers, structure constants etc. My question is whether other things that also pop up all the time in Lie groups, as the (quadratic) Casimir or Dynkin index of an irrep can be visualized this way. (Since that's the only way I can do math :-)

Background: a) In "Birdtracks" an additive formula for Dynkin indices (multiplied with dimensions somehow) of the irreps involved in a Clebsch-Gordan expansion is given, b) I deduced another formula involving writhe normalizers and dimensions that is also valid for any Clebsch-Gordan expansion. (In short: Let $R_i\bigotimes{R_j}=\bigoplus_k{R_k}$, and $W_k=w_i*w_j/w_k$, where the $w_k$ are writhe normalizers, let $o_k$ be the dimensions, then $\Sigma_k{(W_k-W_k^{-1})o_k}=0$. Using pictures the proof is an one-liner.) It's natural to speculate whether a) and b) are essentially the same (or maybe even more things are additive over a Clebsch-Gordan expansion).

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In a), are you referring to (7.24) in "Birdtracks"? –  Bruce Westbury Jul 16 '12 at 12:39
Could you give an example of b)? –  Bruce Westbury Jul 16 '12 at 12:40
Is there any reason to think that there is a relationship other than the observation that both are sums over the composition factors in a tensor product? –  Bruce Westbury Jul 16 '12 at 12:41
@ Bruce Westbury: a) I don't know how many different versions of the ebook float around. I referred to the first table (8.1) in §18 about the E6 family. I assumed (by instinct) for the other families (I worked with E7) similar equations exist. –  Hauke Reddmann Jul 17 '12 at 12:55
@ Bruce Westbury: Regarding your two other questions, I proved that equation "pictorially". I try my best to sketch it in ASCII art: I write >- for a trivalent node and % for a knot crossing. The first one is colored with 3 colors, the latter with 2. %>- = Wijk * >- (structure constants). (By using Reidemeister 3 for nodes, Wijk=wk^2/Wjik.) By Schur, -<>= vanishes (if - and = are different colors). My gauge is setting all theta graphs to 1 which means that if - and = are the same color this graph reduces to - divided by the dimension of the color -, independent of the colors of <>. (contd.) –  Hauke Reddmann Jul 17 '12 at 13:11