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 $π$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, we have an operation $D_{\mathfrak m}(π)$ to construct a simple quotient $τ$ of a Bernstein-Zelevinsky derivative of $π$. This article continues the previous one to study the following poset \[ \mathcal S(π, τ) :=\left\{ \mathfrak n : D_{\mathfrak n}(π)\cong τ\right\} , \] where $\mathfrak n$ runs for all the multisegments. Here the partial ordering on $\mathcal S(π, τ)$ 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.