Multiplicities of weights of some representations and convex polytope (Q1907466)

Let \(G\) be a complex Lie group and let \(Q\) denote the root lattice of \(G\). It is shown that the polynomial \(H_\Lambda(t)\) (\(\Lambda\in Q\) being the highest root) has the property \(\deg H_\Lambda\geq\text{rk }G\) for any simple Lie group and equality holds only for \(A_p\) and \(C_p\). This coincidence suggests the idea that \(H_\Lambda(t)\) can be interpreted in terms of the weight polytope of the adjoint representation of the groups \(A_p\) and \(C_p\). The notion of laminated sequence of polytopes is introduced and all weights for the simple Lie groups are determined.