Plethysm Stability of Schur's $Q$-functions

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.

💡 Research Summary

This paper investigates plethysm stability properties of Schur’s Q‑functions using vertex‑operator techniques, extending the well‑studied stability phenomena for ordinary Schur functions to the spin‑representation setting. After a brief introduction recalling that Schur functions form a basis of the symmetric‑function ring Λ and correspond to irreducible representations of the symmetric group, the authors note that Schur Q‑functions Q_λ provide an analogous basis for the subring Γ⊂Λ associated with projective (spin) representations of the double covering of the symmetric group.

Section 2 establishes the necessary background. Partitions, strict partitions, and the standard statistics ℓ(λ) (length) and |λ| (weight) are defined. The Q‑functions are introduced via a Pfaffian formula Q_λ = Pf M(λ) where the matrix entries involve the bilinear forms Q(r,s) built from the generating series κ_z(A)=∏_{a∈A}(1+za−za). The ring Γ=ℤ