Skew Pieri Rules for Hall-Littlewood Functions

Matjaž Konvalinka,Aaron Lauve

doi:10.46298/dmtcs.3054

Abstract

We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (FPSAC, 2010). The first two were conjectured by the first author (FPSAC, 2011). The key ingredients in the proofs are a q-binomial identity for skew partitions that are horizontal strips and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and antipode. Nous produisons quelques règles dissymètrique de Pieri pour les fonctions Hall–Littlewood au sens de Assaf et McNamara (FPSAC, 2010). Les premières deux règles ont ètè conjecturèe par le premier auteur (FPSAC, 2011). Les principaux ingrèdients dans les preuves sont une identitè q-binomiale pour les partitions dissymètrique qui sont bandes horizontales et une identitè de Hopf qui exprime les produits d'èlèments dissymètrique en termes du coproduit et de l'antipode.

Highlights

Let Λ[t] denote the ring of symmetric functions over Q(t), and let {sλ} and {Pλ(t)} denote its bases of Schur functions and Hall–Littlewood functions, respectively, indexed by partitions λ
We introduce the question via the recent answer for skew Schur functions sλ/μ
We introduce the basics in Subsection 2.1 and return to Λ[t] and Hall–Littlewood functions in Subsection 2.2

Summary

Notation and a key lemma

The conjugate partition of λ is denoted λc. We write mi(λ) for the number of parts of λ equal to i. If λ/μ is not a horizontal strip, define hsλ/μ(t) = 0. If λ/μ is not a vertical strip, define vsλ/μ(t) = 0. For a broken ribbon λ/μ, define brλ/μ(t) = (−t)ht(λ/μ)(1 − t)rib(λ/μ). If λ/μ is not a broken ribbon, define brλ/μ(t) = 0. Lemma 5 For fixed λ, μ, μ ⊆ λ, we have (−t)|λ/ν| vsλ/ν (t) skν/μ(t) = hsλ/μ(t), ν with the sum over all ν, μ ⊆ ν ⊆ λ, for which λ/ν is a vertical strip. We show that if λ/μ is not a horizontal strip and j is the largest index for which λcj − μcj ≥ 2, the term in the product (8) corresponding to j is 0

Elementary Hall–Littlewood identities

Hopf Perspective on Skew Elements

Hopf preliminaries

The Hall–Littlewood setting

Proofs of the main theorems