  Let $F$ be a $p$-adic field. Let $\mathcal{R}$ be the Grothendieck ring of
complex smooth finite-length representations of the groups
$\{GL_n(F)\}_{n=0}^\infty$ taken together, with multiplication defined in the
sense of parabolic induction. We introduce a width invariant for elements of
$\mathcal{R}$ and show that it gives an increasing filtration on the ring.
Irreducible representations of width $1$ are precisely those known as ladder
representations. We thus obtain a necessary condition on irreducible factors of
a product of two ladder representations. For such a product we further
establish a multiplicity-one phenomenon, which was previously observed in
special cases.
