Let $\mathfrak{g}$ be a semisimple Lie algebra with a Cartan subalgebra $\mathfrak{h}^*$ and Weyl group $W$. For $\mu \in \mathfrak{h}^*$ denote by {$\mu$} the unique dominant weight which is $W$-conjugate to $\mu$. Also, let $\rho$ be the half-sum of positive roots and $P(V(\lambda))$ be the set of weights of the irreducible representation defined by a dominant weight $\lambda.$

Now consider two dominant weights $\lambda\neq\lambda'$ and let $\mu \in P(V(\lambda))$. When is the weight $\{\mu + \lambda' + \rho\}- \rho$ not of the form $ k \lambda' + \rho $, for $k$ an integer?