Factorization of certain Macdonald Littlewood-Richardson coefficients

TL;DR

This paper proves a specific factorization formula for certain Macdonald Littlewood-Richardson coefficients, addressing a special case of a broader conjecture relating to when these coefficients equal one.

Contribution

It establishes and proves a factorization formula for Macdonald Littlewood-Richardson coefficients in the case where the associated Kostka number is one, advancing understanding of Stanley's conjecture.

Findings

Proved a factorization formula for specific Macdonald Littlewood-Richardson coefficients.

Confirmed the conjecture for the case when the Kostka number is one.

Contributed to the broader understanding of Stanley's conjecture.

Abstract

We find and prove a factorization formula for certain Macdonald Littlewood-Richardson coefficients $c_{\lambda\mu}^{\nu}(q,t)$. Namely, we consider the case that the Kostka number $K_{\mu, \nu -\lambda}$ is $1$. This settles a particular case of a more general conjecture of Richard Stanely. This conjecture proposes that a factorization formula exists whenever the corresponding regular Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is $1$.