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The braid group surjects onto G_2 tensor space


Let V be the 7-dimensional irreducible representation of the quantum group Uq(g2). For each n, there is a map from the braid group ℬn 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.

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