Hall-Littlewood expansions of Schur delta operators at $t = 0$

Abstract

For any Schur function $s_{\nu}$, the associated {\em delta operator} $\Delta'_{s_{\nu}}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $\nu = (1^{n-1})$ is a column of length $n-1$, the symmetric function $\Delta'_{e_{n-1}} e_n$ appears in the Shuffle Theorem of Carlsson-Mellit. More generally, when $\nu = (1^{k-1})$ is any column the polynomial $\Delta'_{e_{k-1}} e_n$ is the symmetric function side of the Delta Conjecture of Haglund-Remmel-Wilson. We give an expansion of $\omega \Delta'_{s_{\nu}} e_n$ at $t = 0$ in the dual Hall-Littlewood basis for any partition $\nu$. The Delta Conjecture at $t = 0$ was recently proven by Garsia-Haglund-Remmel-Yoo; our methods give a new proof of this result. We give an algebraic interpretation of $\omega \Delta'_{s_{\nu}} e_n$ at $t = 0$ in terms of a $\mathrm{Hom}$-space.