# ALGEBRAIC COMBINATORICS

Pushing our way from the valley Delta to the generalised valley Delta
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 715-725.

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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.239
Classification: 05E05
Keywords: Delta conjecture, Theta operators, Macdonald polynomials
Iraci, Alessandro 1; Vanden Wyngaerd, Anna 2

1 Université du Québec à Montréal Laboratoire d’Algèbre, de Combinatoire et d’Informatique Mathématique (LACIM) 201 Av. du Président-Kennedy Montréal QC H2X 3Y7 (Canada)
2 Université de Paris Institut de Recherche en Informatique Fondamentale (IRIF) Bâtiment Sophie Germain, Case courrier 7014 8 Place Aurélie Nemours Paris 75205 Cedex 13 (France)
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Iraci, Alessandro; Vanden Wyngaerd, Anna. Pushing our way from the valley Delta to the generalised valley Delta. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 715-725. doi : 10.5802/alco.239. https://alco.centre-mersenne.org/articles/10.5802/alco.239/

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