Plethysm Stability of Schur's $Q$-functions

TL;DR

The paper extends plethysm stability from Schur functions to Schur's $Q$-functions by adapting Carré–Thibon's vertex-operator approach to twisted operators. It proves stability for plethysm sequences $(Q_oldsymbol{ u}\circ Q_{p\boldsymbol{ ho}},Q_{s\nu})$ and $(Q_{p\boldsymbol{ u}}\circ Q_{\boldsymbol{ ho}},Q_{s\nu})$ for large $p$ when $ ext{l}(oldsymbol{ ho})>1$, but finds a striking exception where $Q_{(p)}\circ Q_{(m)}$ yields linear growth in $p$, exemplified by $Q_{(p)}(Q_{(m)})=4p z^{pm}$. The work also derives a Schur-like recurrence for $q_n\circ q_m$ via the operator $D_k$, with multiple specializations recovering Butler–King and Murnaghan-type formulas in the $Q$-function setting, and discusses maximal-first-part expansions. Together these results push forward a coherent plethysm theory for Schur's $Q$-functions, with implications for Hall–Littlewood generalizations and connections to affine Lie algebra realizations.

Abstract

Schur functions has been shown to satisfy certain plethysm stability properties and recurrence relations. In this paper, use vertex operator methods to study analogous stability properties of Schur's $Q$-functions. Although the two functions have similar stability properties, we find a special case where the plethysm of Schur's $Q$-functions exhibits linear increase.

Theorems & Definitions (39)

• Proposition 2.1: Sum Rule
• proof
• Corollary 2.2: Skew Sum Rule
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• Proposition 2.6