UC Berkeley

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g₂). For each n, there is a map from the braid group ℬ_n to the endomorphism algebra of the n-th tensor power of V, given by ℛ matrices. Extending linearly to the braid group algebra, we get a map 

AB_n --> End_{U_q(g_2)}(V^{\otimes n}).

Lehrer and Zhang have proved that map is surjective, as a special case of a more general result.

Using Kuperberg’s spider for G₂, we give an elementary diagrammatic proof of this result.