"Pushing" our way from the valley Delta to the generalised valley Delta

TL;DR

This paper demonstrates that the valley version of the Delta conjecture is equivalent to its generalized form and shows that it implies the generalized Delta square conjecture, advancing understanding of symmetric function identities.

Contribution

The authors prove the equivalence between the valley version and the generalized Delta conjecture using pushing algorithms and symmetric function identities.

Findings

Valley Delta conjecture implies the generalized Delta conjecture.

Valley Delta conjecture implies the generalized Delta square conjecture.

The equivalence of valley and generalized versions is established.

Abstract

In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so called Delta conjecture, the rise version and the valley version. Of the former, they also give a more general statement in which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley version is also formulated. In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove the generalised version of the shuffle theorem. An extension of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley version of the (generalised) Delta square conjecture, and to suggest a symmetric function identity later stated and proved in [D'Adderio, Romero 2020]. In this paper, we use the pushing algorithm together with the aforementioned symmetric function identity in order to prove that the valley version of the Delta conjecture implies the valley version of the generalised Delta conjecture, which means that they are actually equivalent. Combining this with the results in [Iraci, Vanden Wyngaerd 2020], we prove that the valley version of the Delta conjecture also implies the corresponding generalised Delta square conjecture.