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In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials [A. M. Garsia and M. Haiman. A remarkable q, t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191-244], [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002), 371-407]. The Delta Conjecture involves two parameters q, t; in this article we give the first proof that the Delta Conjecture is true when q = 0 or t = 0.

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A Proof of the Delta Conjecture When $$\varvec{q=0}$$

Garsia

Haglund

Remmel

et al. 2019

Ann. Comb.

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A Valley Version of the Delta Square Conjecture

Iraci

Wyngaerd

2021

Ann. Comb.

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Inspired by [Qiu, Wilson 2020] and [D'Adderio, Iraci, Vanden Wyngaerd 2019], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the Θ f operators introduced in [D'Adderio, Iraci, Vanden Wyngaerd 2020-Theta Operators].

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Some Consequences of the Valley Delta Conjectures

D’Adderio,

Iraci

2023

Ann. Comb.

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Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $$\Delta '_{e_{n-k-1}} e_n$$ Δ e n - k - 1 ′ e n in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).

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The generalized Delta conjecture at t=0

Cited by 4 publications

References 14 publications

A Proof of the Delta Conjecture When $$\varvec{q=0}$$

A Proof of the Delta Conjecture When $$\varvec{q=0}$$

A Valley Version of the Delta Square Conjecture

Some Consequences of the Valley Delta Conjectures

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