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Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments III: properties of minimal sequences

Let $F$ be a non-Archimedean local field. For an irreducible smooth representation $π$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, one associates a simple quotient $D_{\mathfrak m}(π)$ of a Bernstein-Zelevinsky derivative of $π$. In the preceding article, we showed that \[ \mathcal S(π, τ) :=\left\{ \mathfrak m : D_{\mathfrak m}(π)\cong τ\right\} , \] has a unique minimal element under the Zelevinsky ordering, where $\mathfrak m$ runs for all multisegments. The main result of this article includes commutativity and subsequent property of the minimal sequence. At the end of this article, we conjecture some module structure arising from the minimality.