Haglund's positivity conjecture for multiplicity one pairs

TL;DR

This paper proves Haglund's positivity conjecture for certain pairs of partitions where the Kostka number is one, and explores the transition matrix between Macdonald and Schur functions.

Contribution

It establishes the conjecture for pairs with Kostka number one and provides new insights into the transition matrix between Macdonald and Schur functions.

Findings

Proves Haglund's conjecture for pairs with Kostka number one.

Provides results on the transition matrix between Macdonald and Schur functions.

Enhances understanding of positivity properties in symmetric functions.

Abstract

Haglund's conjecture states that $\dfrac{\langle J_{\lambda}(q,q^k),s_\mu \rangle}{(1-q)^{|\lambda|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $\lambda,\mu$ and all non-negative integers $k$, where $J_{\lambda}$ is the integral form Macdonald symmetric function and $s_\mu$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(\lambda,\mu)$ satisfies $K_{\lambda,\mu}=1$ or $K_{\mu',\lambda'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.