Giambelli, Pieri, and tableau formulas via raising operators

TL;DR

This paper presents a new proof connecting Giambelli and Pieri formulas for Hall-Littlewood functions using raising operators, and develops tableau formulas and recursions for related symmetric functions and Schubert classes.

Contribution

It introduces a raising operator approach to prove formula equivalences and derive tableau formulas for Hall-Littlewood functions and related symmetric functions.

Findings

Proved the equivalence of Giambelli and Pieri formulas using raising operators.

Derived new recursions for Schubert classes.

Established tableau formulas for Hall-Littlewood functions and Stanley symmetric functions.

Abstract

We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew Young diagram.