Y\right)}$. Using it to identify $\mathfrak t$ and $\mathfrak t^*$, a positive Weyl chamber $\mathfrak t^*_+$ is given by the set of diagonal matrices $\lambda = \operatorname{diag}(\lambda_1, \ldots, \lambda_d)$ with purely imaginary entries, summing to zero and arranged in such a way that $i \lambda_1 \geq \ldots \geq i \lambda_d$. This corresponds to choosing the positive roots $\alpha_{j,k}(\lambda) = i(\lambda_j - \lambda_k)$ with $j < k$. The points in the interior $\mathfrak t^*_{>0}$ are those $\lambda \in \mathfrak t^*_+$ with all distinct eigenvalues, i.e., $i \lambda_1 > \ldots > i \lambda_d$.

Let $M$ be a compact, connected Hamiltonian $K$-manifold of dimension $2n$, with symplectic form $\omega_M$ and a choice of moment map $\Phi_K \colon M \rightarrow \mathfrak k^*$ [@cannasdasilva08; @guilleminsternberg84]. The intersection $\Delta_K(M) = \Phi_K(M) \cap \mathfrak t^*_+$ of its image with the positive Weyl chamber is a compact convex polytope, called the *moment polytope* or *Kirwan polytope* [@guilleminsternberg82; @kirwan84b; @guilleminsjamaar05]. If $M$ is a coadjoint $K$-orbit $\mathcal O_\lambda$ through some $\lambda \in \mathfrak t^*_+$, it will always be equipped with the Kirillov–Kostant–Souriau symplectic form and the moment map induced by the inclusion $\mathcal O_\lambda \subseteq \mathfrak k^*$. Evidently, $\Delta_K(\mathcal O_\lambda) = \{\lambda\}$.

Throughout this article, we will always impose 