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

Abstract

For any Schur function $s_{\nu }$, the associated {\em delta operator} ${\Delta }_{s_{\nu }}^{\prime }$ 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 ${\Delta }_{e_{n-1}}^{\prime }{e}_n$ appears in the Shuffle Theorem of Carlsson-Mellit. More generally, when $\nu =(1^{k-1})$ is any column the polynomial ${\Delta }_{e_{k-1}}^{\prime }{e}_n$ is the symmetric function side of the Delta Conjecture of Haglund-Remmel-Wilson. We give an expansion of $\omega {\Delta }_{s_{\nu }}^{\prime }{e}_n$ 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 $\omega {\Delta }_{s_{\nu }}^{\prime }{e}_n$ at $t=0$ in terms of a $\mathrm{Hom}$-space.

arXiv comment18 pages

References in corpus (2)

• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• A proof of the Delta Conjecture when $q=0$

Cited by in corpus (7)

• A proof of the Delta Conjecture when $q=0$
• The Hilbert series of the superspace coinvariant ring
• The Delta square conjecture
• The generalized Delta conjecture at t=0
• (GL_k x S_n)-modules and nabla of hook-indexed Schur functions
• Harmonic bases for generalized coinvariant algebras
• The valley version of the Extended Delta Conjecture