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

Qiu Dun, Jeffrey B. Remmel, Emily Sergel, Guoce Xin

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.


Keywords


Schur positivity, Macdonald polynomials, Delta operator

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