In this chapter we classify and construct all invariance groups whose primitive invariant tensors are a symmetric bilinear*d*_{ab}, and a symmetric trilinear*d*_{abc}, satisfying the relation (19.16).

Take as primitives a symmetric quadratic invariant*d*_{ab}and a symmetric cubic invariant*d*_{abc}. As explained in chapter 12, we can use*d*_{ab}to lower all indices. In the birdtrack notation, we drop the open circles denoting symmetric 2-index invariant tensor*d*^{ab}, and we drop arrows on all lines:

The defining*n*-dimensional rep is by assumption irreducible, so

Were (19.3) nonvanishing, we could use to project out a 1-dimensional subspace, violating the

EP - 217 PB - Princeton University Press PY - 2008 SN - 9780691118369 SP - 210 T2 - Group Theory T3 - Birdtracks, Lie's, and Exceptional Groups UR - http://www.jstor.org/stable/j.ctt7rnjq.22 Y2 - 2020/09/19/ ER -