SIMPLE-CURRENT SYMMETRIES, RANK LEVEL DUALITY, AND LINEAR SKEIN RELATIONS FOR CHERN-SIMONS GRAPHS

A previously proposed two-step algorithm for calculating the expectati on values of arbitrary Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solvi ng certain non-linear equations is repaired by introducing additional linear equations. The step which involves reducing arbitrary graphs to sums of products of tetrahedra remains seriously disabled, apart from a few exceptional cases. As a first step towards a new algorithm for general graphs we find useful linear equations for those special graph s which support knots and links. Using the improved set of equations f or tetrahedra we examine the symmetries between tetrahedra generated b y arbitrary simple currents. Along the way we describe the simple, cla ssical origin of simple-current charges. The improved skein relations also lead to exact identities between planar tetrahedra in level K G(N ) and level N G(K) Chern-Simons theories, where G(N) denotes a classic al group. These results are recast as WZW braid-matrix identities and as identities between quantum 6j-symbols at appropriate roots of unity . We also obtain the transformation properties of arbitrary graphs, kn ots, and links under simple-current symmetries and rank-level duality. For links with knotted components this requires precise control of th e braid eigenvalue permutation signs, which we obtain from plethysm an d an explicit expression for the (multiplicity-free) signs, valid for all compact gauge groups and all fusion products.