The generalized Delta conjecture at t=0

TL;DR

This paper proves specific cases (q=0 and t=0) of the generalized Delta conjecture involving symmetric functions, extending recent results and also confirming part of a related conjecture.

Contribution

It establishes the q=0 and t=0 cases of the generalized Delta conjecture, advancing the understanding of symmetric functions in algebraic combinatorics.

Findings

Proved the q=0 and t=0 cases of the generalized Delta conjecture.

Extended recent results by Garsia, Haglund, Remmel, and Yoo.

Confirmed the q=0 case of the generalized Delta square conjecture.

Abstract

We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.