The braid group surjects onto $G_2$ tensor space

TL;DR

This paper provides an elementary diagrammatic proof that the braid group surjects onto the endomorphism algebra of the tensor powers of the 7-dimensional G_2 representation, confirming a key algebraic surjectivity result.

Contribution

It offers a new, diagrammatic proof of the surjectivity of the braid group onto the G_2 tensor space endomorphisms, simplifying previous algebraic approaches.

Findings

Confirmed the surjectivity of the braid group onto G_2 tensor space endomorphisms.

Provided an elementary diagrammatic proof using Kuperberg's G_2 spider.

Simplified understanding of the algebraic structure related to G_2 representations.

Abstract

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). 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. We can extend this linearly to a map on the braid group algebra. Lehrer and Zhang (MR2271576) prove this map is surjective, as a special case of a more general result. Using Kuperberg's spider for G_2 from arXiv:math.QA/9201302, we give an elementary diagrammatic proof of this result.