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

TL;DR

This paper provides a new expansion of certain symmetric functions related to delta operators at t=0, offering a new proof of the Delta Conjecture at t=0 and an algebraic interpretation involving Hom-spaces.

Contribution

It introduces a novel expansion of omega Delta'_{s_{nu}} e_n at t=0 in the dual Hall-Littlewood basis for any partition nu, and offers a new proof of the Delta Conjecture at t=0.

Findings

Expansion of omega Delta'_{s_{nu}} e_n at t=0 in dual Hall-Littlewood basis

New proof of the Delta Conjecture at t=0

Algebraic interpretation via Hom-spaces

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.