  Path algebras are a convenient way of describing decompositions of tensor
powers of an object in a tensor category. If the category is braided, one
obtains representations of the braid groups $B_n$ for all $n\in \N$. We say
that such representations are rigid if they are determined by the path algebra
and the representations of $B_2$. We show that besides the known classical
cases also the braid representations for the path algebra for the 7-dimensional
representation of $G_2$ satisfies the rigidity condition, provided $B_3$
generates $\End(V^{\otimes 3})$. We obtain a complete classification of ribbon
tensor categories with the fusion rules of $\g(G_2)$ if this condition is
satisfied.
