Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 3 / i / Solution

The ambient group ring of a weight lattice isAddition is coefficientwise, and multiplication uses addition in the weight lattice. For a finite-dimensional Lie algebra representation $V={⨁}_{\lambda }{V}_{\lambda }$, its formal character isThe formal character records the dimensions of all weight spaces. The ambient lattice ring should be distinguished from the representation ring of a semisimple Lie algebra, which identifies with its Weyl-invariant subring.

For the ${\mathfrak{sl}}_3$ module ${\Gamma }_{1,2}$, $L_1+L_2+L_3=0$ and the highest weight is $L_1-2L_3={\omega }_1+2{\omega }_2$. The decomposition $V\otimes {\mathrm{Sym}}^2{V}^{\ast }={\Gamma }_{1,2}\oplus V^{\ast }$ givesCollecting equal weights yieldsThe six first weights and three second weights have multiplicity one; each of the three weights $-L_i$ has multiplicity two. Their total is $6+3+6=15$, as required.