On the Schur positivity of $\Delta_{e_2} e_n[X]$

TL;DR

This paper proves that for the case k=2, the coefficients of Schur functions in _{e_2} e_n[X] are polynomials with non-negative coefficients in q and t, confirming a conjecture about Schur positivity.

Contribution

The authors provide four proofs demonstrating that the coefficients are positive polynomials in q and t for the case k=2, strengthening previous conjectures.

Findings

Coefficients have positive q,t-analog expansions.

Four different proofs establish Schur positivity for k=2.

Supports conjecture on polynomial positivity in symmetric function expansions.

Abstract

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.