Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 5 / 2 / c / Solution

Every finite-dimensional irreducible Lie algebra representation has highest weight $\lambda =a{\Lambda }_1+b{\Lambda }_2$ for unique integers $a,b\ge 0$. The Weyl vector isso $\lambda +\rho =(a+b/2+3/2,b/2+1/2)$. In the Weyl dimension formula, the factors for the four positive roots, ordered as ${\epsilon }_1-{\epsilon }_2,{\epsilon }_2,{\epsilon }_1,{\epsilon }_1+{\epsilon }_2$, are respectivelyHenceIn particular the two fundamental representations have dimensions five and four. The highest root is $\theta ={\epsilon }_1+{\epsilon }_2=2{\Lambda }_2$, so the Adjoint representation has dimension ten. These also fix the dimensions of the crystals below. Our numbering places the long root first; interchanging a B2/C2 numbering without interchanging the associated fundamental-weight labels would give the wrong dimension formula.