For integral weights $λ$ and $μ$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant's weight multiplicity formula gives the multiplicity of the weight $μ$ in the irreducible representation with highest weight $λ$, which we denote by $m(λ,μ)$. Kostant's weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(λ,μ)$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(λ,μ)$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra $\mathfrak{sl}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tildeα,μ)$, where $\tildeα$ is the highest root and $μ$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry's conjecture that the $q$-analog of Kostant's weight multiplicity formula is $m_q(\tildeα,μ)=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $μ=-(α_i+α_{i+1}+\cdots+α_{j})$ is a negative root of $\mathfrak{sl}_{r+1}(\mathbb{C})$.