The braid group surjects onto G₂ tensor space

Scott Morrison

Vol. 249 (2011), No. 1, 189–198

DOI: 10.2140/pjm.2011.249.189

Abstract

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

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.

Keywords

braid group, spider, G2, tensor category, representation theory

Mathematical Subject Classification 2000

Primary: 17B37

Secondary: 20G42, 17B10, 18D10

Supplementary material

Mathematica notebook

Milestones

Received: 22 September 2009

Revised: 4 January 2010

Accepted: 4 January 2010

Published: 3 January 2011