A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials

TL;DR

This paper proves a special case of Stanley's conjecture relating Littlewood-Richardson coefficients for Jack and Macdonald polynomials, showing they can be expressed as products of weighted hooks for partitions with up to three parts.

Contribution

It establishes a product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials when partitions have at most three parts, extending Stanley's conjecture.

Findings

Proved the conjecture for partitions with up to 3 parts.

Extended the result from Jack to Macdonald polynomials.

Demonstrated the product formula for specific Littlewood-Richardson coefficients.

Abstract

Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson coefficient for a triple of Schur polynomials is 1, then the corresponding coefficient for Jack polynomials can be expressed as a product of weighted hooks of the Young diagrams associated to the partitions indexing the coefficient. We prove a special case of this conjecture in which the partitions indexing the Littlewood-Richardson coefficient have at most 3 parts. We also show that this result extends to Macdonald polynomials.