  Let $F$ be a non-Archimedean field. A sequence of derivatives of generalized
Steinberg representations can be used to construct simple quotients of
Bernstein-Zelevinsky derivatives of irreducible representations of
$\mathrm{GL}_n(F)$. We introduce a notion of a highest derivative multisegment
and use it to give a combinatorial criteria when two sequences of derivatives
of generalized Steinberg representations give rise isomorphic simple quotients
of Bernstein-Zelevinsky derivatives of an irreducible representation. We also
show a certain uniqueness statement which hence gives a canonical sequence and
show that such sequence possesses several useful properties. Those results can
be translated to affine Hecke algebra of type A, by using our previous
formulation of Bernstein-Zelevinsky derivatives in affine Hecke algebra of type
A setting.

Representation Theory;Number Theory