  The inner plethysm of symmetric functions corresponds to the $\lambda$-ring
operations of the representation ring $R({\mathfrak S}_n)$ of the symmetric
group. It is known since the work of Littlewood that this operation possesses
stability properties w.r.t. $n$. These properties have been explained in terms
of vertex operators [Scharf and Thibon, Adv. Math. 104 (1994), 30-58]. Another
approach [Orellana and Zabrocki, Adv. Math. 390 (2021), \# 107943], based on an
expression of character values as symmetric functions of the eigenvalues of
permutation matrices, has been proposed recently. This note develops the theory
from scratch, discusses the link between both approaches and provides new
proofs of some recent results.
