# 6j symbols of SU(4) at level 4

Does anybody know of a reference that gives the (quantum) 6j symbols of SU(4) at level 4?

Alternatively, I know the S-matrix and the fusion rules, in the form $a \times b = \sum_i N^{ab}_{c_i} c_i$ Is there a simple way to compute all quantum 6j symbols from these?

• Welcome to Mathoverflow. You can use latex to improve your posts. – Chris Ramsey Oct 10 '15 at 14:27
• This question is a little unclear as it stands. can you include definitions of $N$ and $a,b,c$? This will help uses from pure math decipher applied questions better. – Benjamin Oct 10 '15 at 16:18
• The $N_{ab}^c$ are the dimensions of the $hom(a \otimes b,c )$ spaces in the fusion categories for $SU(4)_4$. – Matthew Titsworth Oct 10 '15 at 22:10
• I can't think of a reference for your first question and I don't think there is one. All of the sources I can think of are for $SU(2)_k$. The standard knowledge about solving pentagon equations however is basically contained in this answer. – Matthew Titsworth Oct 10 '15 at 22:35
• I think there are some physics papers that calculate part of the 6j symbols for $SU(N)$. But, to me it is actually not even clear if there is a unique categorification of the $SU(N)_k$ fusion rules for $N>2$. – Marcel Bischoff Oct 13 '15 at 14:34

In answer to your second question, no, there is not a simple way to compute $6j$-symbols from the fusion rules and the $S$-matrix.