Haglund's conjecture on 3-column Macdonald polynomials

TL;DR

This paper proves a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by 3-tuple skew shapes, confirming Haglund's conjecture and providing new insights into Macdonald polynomials.

Contribution

It introduces a novel positive combinatorial formula for certain LLT and Macdonald polynomials, verifying a longstanding conjecture of Haglund.

Findings

Verified Haglund's conjecture for 3-column Macdonald polynomials

Derived a positive combinatorial rule for transformed Macdonald polynomials

Expressed noncommutative Schur functions as positive sums in Lam's algebra

Abstract

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Macdonald polynomials indexed by a shape with 3 columns.