Let $I_1(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_1$ and representation $\lambda_1$ having the S matrix $S_1$. Let $I_2(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_2$ and representation $\lambda_2$ having the S matrix $S_2$ (hooray for copy'n'paste!). Then, for now just by my experiments, there always exists an invariant $I(L)=I_1(L)*I_2(L)$ (good old multiplication) with the S matrix $S_1\otimes{S_2}$ (here the trouble already begins: you can have S in tensor from, or condensed to matrix form, or condensed with index swap - and "$\otimes$" may be correct only in one case) which surely comes from the group $G_1°G_2$ (Tensor product? Direct product? Product of my imagination? :-) and the representation $\lambda_1°\lambda_2$ (even worse - if I remember correctly, now there are even 3 kind of products to be considered).

Please help me out with the correct signs $*,\otimes,\times$ which I confuse all the time. I also like to know about sort of a converse: If $G=G_1°G_2$ (Tensor? Direct?), are the representations $\lambda$ of $G$ always of the form $\lambda_1°\lambda_2$?

Algebra), then moving on to books on the representation theory of finite groups (I like James and Liebeck'sRepresentations and Characters of Groups) before tackling the harder stuff. – Qiaochu Yuan Aug 28 '11 at 4:42