Raising operators and the Littlewood-Richardson polynomials

TL;DR

This paper introduces a new Pieri rule for double symmetric functions using Young's raising operators, providing a novel proof for calculating Littlewood--Richardson polynomials and advancing the understanding of double Schur functions.

Contribution

It develops a Pieri rule for the ring of double symmetric functions via raising operators, leading to a new proof for Littlewood--Richardson polynomial calculations.

Findings

Derived a Pieri rule for double symmetric functions.

Provided a new proof for Littlewood--Richardson polynomial calculation.

Connected raising operators with the structure of double Schur functions.

Abstract

We use Young's raising operators to derive a Pieri rule for the ring generated by the indeterminates $h_{r,s}$ given in Macdonald's 9th Variation of the Schur functions. Under an appropriate specialisation of $h_{r,s}$, we derive the Pieri rule for the ring $\La(a)$ of double symmetric functions, which has a basis consisting of the double Schur functions. Together with a suitable interpretation of the Jacobi--Trudi identity, our Pieri rule allows us to obtain a new proof of a rule to calculate the Littlewood--Richardson polynomials, which gives a multiplication rule for the double Schur functions.