Cyclic representations of general linear p-adic groups

Let $\pi_1,\ldots,\pi_k$ be smooth irreducible representations of $p$-adic general linear groups. We prove that the parabolic induction product $\pi_1\times\cdots\times \pi_k$ has a unique irreducible quotient whose Langlands parameter is the sum of the parameters of all factors (cyclicity property), assuming that the same property holds for each of the products $\pi_i\times \pi_j$ ($i<j$), and that for all but at most two representations $\pi_i\times \pi_i$ remains irreducible (square-irreducibility property). Our technique applies the recently devised Kashiwara-Kim notion of a normal sequence of modules for quiver Hecke algebras. Thus, a general cyclicity problem is reduced to the recent Lapid-M\'inguez conjectures on the maximal parabolic case.