Stability properties of inner plethyms (Lecture Notes)

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.