Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 1

The squared root lengths and inner product areWith $A_{ij}=2{\alpha }_i\cdot {\alpha }_j/|{\alpha }_j{|}^2$, the Cartan matrix isThis is the Cartan matrix of the G2 root system, with ${\alpha }_1$ short and ${\alpha }_2$ long.

The root string through ${\alpha }_1$ in the ${\alpha }_2$ direction contains ${\alpha }_1$ and ${\alpha }_1+{\alpha }_2$, so ${\alpha }_1$ can be raised once by ${\alpha }_2$. The string through ${\alpha }_2$ in the ${\alpha }_1$ direction containsso ${\alpha }_2$ can be raised three times by ${\alpha }_1$.

The Weyl reflection in the hyperplane perpendicular to $\alpha$ sends a weight $\mu$ to

The positive roots generated by the two simple roots areThe Adjoint representation of a Lie algebra has one weight space for each positive and negative root and a rank-two zero-weight space. Its nonzero weights are thereforetogether with their negatives. Thus the diagram consists of a hexagon of six long roots, a hexagon of six short roots, and the origin with multiplicity two. The representation has dimension $12+2=14$.

The fundamental weights satisfySolving these equations gives

Starting from the highest weight ${\mu }_1=2{\alpha }_1+{\alpha }_2$, lowering with the simple-root operators and closing under the Weyl group givesEvery weight has multiplicity one. The diagram is the hexagon of short roots with one central weight, so this is the seven-dimensional fundamental irreducible representation of $G_2$.