Stability properties of inner plethyms (Lecture Notes)

TL;DR

This paper explores the stability properties of inner plethysms in symmetric functions, connecting different theoretical approaches and offering new proofs of recent results in the representation theory of symmetric groups.

Contribution

It develops the theory of inner plethysm stability from scratch, links existing approaches, and provides new proofs of recent findings.

Findings

Clarifies the connection between vertex operator approach and eigenvalue-based approach

Provides new proofs of stability properties of inner plethysm

Enhances understanding of symmetric group representation ring operations

Abstract

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.