Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups

Question

Which $6j$-symbols for quantised enveloping algebras are known explicitly?

The quantum $6j$-symbols for $sl(2)$ are well-known. The references are Masbaum and Vogel and Frenkel and Khovanov.

What is known for other simple Lie algebras?

In case this seems somewhat vague here is a precise question. The data for a $6j$-symbol starts with a tetrahedral graph with edges labelled by highest weights. Then usually there is additional information needed at the vertices which I want to avoid. Take the example of $sl(n)$ and use partitions instead of highest weights. Label two opposite edges by a partition of the form $1^k$ (corresponding to an exterior power of the vector representation) and label the other four edges by partitions. Then associated to this data is a scalar.

Then I would expect this function to be determined by linear recurrence relations (i.e. D-finite or holonomic). Is this correct? and if so can you give recurrence relations?

Ideally we would also regard $n$ as an indeterminate.

Background In general $6j$-symbols arise for any semisimple abelian category which is also monoidal. They are the components of the natural transformation $(\otimes)(\otimes \times 1)\cong (\otimes)(1\times \otimes)$.

In more down to earth terms. If you know the Grothendieck ring of a semisimple abelian monoidal category and you attempt to construct this then the information you are missing is the $6j$-symbols. You can construct the abelian category and you can construct the tensor product functor but you don't have the associator.

For example it is well-known that the character table of a finite group does not determine the group. It does determine the category of representations as a semisimple abelian category. The $6j$-symbols are needed to make this a monoidal category.

For further discussion see:
Character table does not determine group Vs Tannaka duality

Answer 1

We can also use partitions ($k$) (symmetric powers) instead of ($1^k$) on one or two of the edges. This still gives just scalars, but includes the full story for sl(2).

This problem seems to be equivalent to the problem of computing the exchange operator in the tensor product of two (quantum) symmetric or exterior powers of the vector representation of the quantum sl(n), e.g. $S^kV\otimes S^mV$, in the standard basis of the symmetric (resp., exterior) powers (about exchange operators see my paper with Varchenko arXiv:math/9801135 and my ICM talk arXiv:math/0207008). This can be computed if you know the fusion operator for these representations, which can be computed efficiently using the ABRR (Arnaudon-Buffenoir-Ragoucy-Roche) equation, see e.g. the appendix to arXiv:math/9801135. I am not sure if the answer is completely worked out anywhere, but there are at least some answers. For instance, see the paper arXiv:q-alg/9704005 where something is done even in the elliptic setting (which relates to elliptic 6j symbols of Frenkel-Turaev). What they do is compute the matrix elements for $m=1$, but the general $m$ can be obtained using the fusion procedure. This should be a really nice computation with a nice answer of the type you are expecting. In particular, in a special case you'll get coefficients of Macdonald's difference operators attached to symmetric powers. In the exterior powers case (or a product of a symmetric and an exterior power) the answer will be simpler, since $k$ cannot get larger than $n$ (in this case you should perhaps get a pure product, and it should be completely covered in the above paper).

EDIT: Remark. Let $V,W$ be representations of the quantum group with 1-dimensional zero weight spaces. Then a natural basis in ${\rm Hom}(L_\lambda, (V\otimes L_\mu)\otimes W)$ is the compositions of intertwiners in one order, while a natural basis of ${\rm Hom}(L_\lambda, V\otimes (L_\mu\otimes W))$ is the composition of intertwiners in the other order. Thus, the 6j-symbol matrix (which is by definition the transition matrix between these two bases) is the exchange operator for intertwiners.