Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 102 / 5 / c / Solution

The weights of the seven-dimensional irreducible $G_2$ representation are zero and the six short roots. In the $A_2$ weight lattice, the three positive short roots areTogether with their negatives, they split into the weight sets of the two dual three-dimensional Fundamental representations of sl3; the zero weight supplies a trivial representation. By the Restriction of the seven-dimensional G2 representation to long-root A2,