AI 中文总结
研究对称群$S_n$中 Verma 重数满足$[M(p):L(q)] \geq 2$的置换，通过构造和图示展示其生成$S_{n + 1}$中同性质置换的方法，还给出了$S_5$、$S_6$和$S_7$中相关置换的情况。
AI 中文摘要
我们考虑对称群$S_n$中的置换$q$，其在$\mathcal{O}(\mathfrak{sl}_n)$主块中的 Verma 重数满足$[M(p):L(q)] \geq 2$。我们给出一种构造及图示可视化，展示具有此性质的$S_n$中的置换如何生成$S_{n + 1}$中具有相同重数性质的一族置换。虽该方法不能找回$S_{n + 1}$中所有此类置换，但能系统地生成许多新例子。此外，我们在 Bruhat 图中给出$S_5$中满足$[M(\mathrm{id}):L(q)] \geq 2$的所有置换，并描述$S_6$和$S_7$中具有非简单 Verma 重数的所有置换。
英文摘要
We consider permutations $q$ in the symmetric group $S_n$ whose Verma multiplicities in the principal block of $\mathcal{O}(\mathfrak{sl}_n)$ satisfy $[M(p):L(q)] \geq 2$. We present a construction along with a diagrammatic visualization, showing how permutations in $S_n$ with this property generate a family of permutations in $S_{n+1}$ that share the same multiplicity property. While the method does not recover all such permutations in $S_{n+1}$, it systematically generates many new examples. In addition, we present all permutations in $S_5$ with $[M(\mathrm{id}):L(q)] \geq 2$ in a Bruhat diagram and describe all permutations in $S_6$ and $S_7$ with non-simple Verma multiplicities.
