Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments II: Minimal sequences

Let $F$ be a non-Archimedean local field. For any irreducible smooth representation $\pi$ of ${\mathrm{G}\mathrm{L}}_n(F)$ and a multisegment $\mathfrak{m}$, we have an operation $D_{\mathfrak{m}}(\pi )$ to construct a simple quotient $\tau$ of a Bernstein-Zelevinsky derivative of $\pi$. This article continues the previous one to study the following poset $$\mathcal{S}(\pi ,\tau ):=\{\mathfrak{n}:D_{\mathfrak{n}}(\pi )\cong \tau \},$$ where $\mathfrak{n}$ runs for all the multisegments. Here the partial ordering on $\mathcal{S}(\pi ,\tau )$ comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.