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

  • Qiu Dun University of California San Diego
  • Jeffrey B. Remmel University of California San Diego
  • Emily Sergel University of Pennsylvania
  • Guoce Xin Capital Normal University
Keywords: Schur positivity, Macdonald polynomials, Delta operator

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.

Published
2018-10-19
Article Number
P4.20