Theta operators, refined Delta conjectures, and coinvariants

D'Adderio, Michele; Iraci, Alessandro; Wyngaerd, Anna Vanden

doi:10.1016/j.aim.2020.107447

Cited by 18 publications

(34 citation statements)

References 18 publications

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“…Drawing terminology from supersymmetry, we will refer to commuting variables as bosonic and anticommuting variables as fermionic. D'Adderio, Iraci, and Vanden Wyngaerd conjectured [3] a generalization of Haiman's result involving two sets of bosonic and two sets of fermionic variables of which Haiman's result forms the 'purely bosonic case'. We prove the 'purely fermionic case' of their conjecture.…”

Section: Introductionmentioning

confidence: 98%

“…The ring DR n is a bigraded S n -module; Haiman used algebraic geometry to calculate its isomorphism type [10]. In recent years, researchers in algebraic combinatorics studied variants of DR n involving mixtures of commuting and anticommuting variables [1,2,3,9,11,12,13,14,16,17,18,19]. Drawing terminology from supersymmetry, we will refer to commuting variables as bosonic and anticommuting variables as fermionic.…”

Section: Introductionmentioning

confidence: 99%

“…over the ground field C(q, t). Given any element F ∈ Λ, D'Adderio, Iraci, and Vanden Wyngaerd introduced [3] a Theta operator Θ F : Λ → Λ whose definition is recalled in Section 2. Building on the techniques in [3], D'Adderio and Mellit [4] used Theta operators to prove the rise version of the Delta Conjecture of Haglund, Remmel, and Wilson [8].…”

Section: Introductionmentioning

confidence: 99%

“…Given any element F ∈ Λ, D'Adderio, Iraci, and Vanden Wyngaerd introduced [3] a Theta operator Θ F : Λ → Λ whose definition is recalled in Section 2. Building on the techniques in [3], D'Adderio and Mellit [4] used Theta operators to prove the rise version of the Delta Conjecture of Haglund, Remmel, and Wilson [8]. D'Adderio and Romero [5] rederived and clarified a slew of symmetric function identities using Theta operators, drastically shortening many of their proofs.…”

Section: Introductionmentioning

confidence: 99%

“…If G ∈ Λ has degree d in the x-variables, then Θ i G has degree d + i and will typically involve the parameters q and t (even if G itself does not). D'Adderio, Iraci, and Vanden Wyngaerd conjectured [3] that symmetric functions of the form Θ i Θ j ∇e n−i−j (where ∇ is the Bergeron-Garsia nabla operator) determine the quadruply-graded S n -structure of T DR n . Deferring various definitions to Section 2, their conjecture may be stated as follows.…”

Section: Introductionmentioning

confidence: 99%

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A proof of the fermionic Theta coinvariant conjecture

Iraci¹,

Rhoades²,

Romero³

2022

Preprint

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Let (x1, . . . , xn, y1, . . . , yn) be a list of 2n commuting variables, (θ1, . . . , θn, ξ1, . . . , ξn) be a list of 2n anticommuting variables, and C[xn, yn] ⊗ ∧{θn, ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded Snisomorphism type of C[xn, yn] ⊗ ∧{θn, ξ n }/I where I is the ideal generated by Sn-invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables equal xi, yi equal to zero.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A proof of the fermionic Theta coinvariant conjecture

Iraci¹,

Rhoades²,

Romero³

2022

Preprint

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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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Theta operators, refined Delta conjectures, and coinvariants

Cited by 18 publications

References 18 publications

A proof of the fermionic Theta coinvariant conjecture

A proof of the fermionic Theta coinvariant conjecture

A Valley Version of the Delta Square Conjecture

Some Consequences of the Valley Delta Conjectures

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