A proof of the Extended Delta Conjecture

Blasiak, Jonah; Haiman, Mark; Morse, Jennifer Q.; Anna, Pun,; Seelinger, George H.

doi:10.48550/arxiv.2102.08815

Cited by 4 publications

(13 citation statements)

References 19 publications

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“…On another related line of research, the Delta Conjecture of Haglund, Remmel, and Wilson [18] predicts two combinatorial formulas for a particular symmetric function ∆ e k−1 e n (q, t) with q and t parameters coming from the theory of Macdonald polynomials. The "rise formula" half of the conjecture has been recently proven independently by D'Adderio and Mellit [8] and by Blasiak, Haiman, Morse, Pun, and Seelinger [1], and the two works prove different generalizations of the rise formula.…”

Section: Hall-littlewood Symmetric Functionmentioning

confidence: 97%

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Springer fibers and the Delta Conjecture at $t=0$

T.¹,

Levinson²,

Woo³

2021

Preprint

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We introduce a family of varieties Y n,λ,s , which we call the ∆-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H * (Y n,λ,s ) and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induced Specht modules. The λ = (1 k ) case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at t = 0. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety Y n,λ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud-Saltman rank variety and diagonal matrices.

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Section: Hall-littlewood Symmetric Functionmentioning

confidence: 97%

“…The operations of deleting a cell, applying the flattening function to the labels, and possibly shifting a row to the left all preserve (S3), so T (i) has property (S3). Since (S4) only concerns ] determined by reading order and the fillings T (1) and T (3) , which are also Schubert compatible.…”

Section: An Affine Paving Of the ∆-Springer Varietymentioning

confidence: 99%

Springer fibers and the Delta Conjecture at $t=0$

T.¹,

Levinson²,

Woo³

2021

Preprint

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“…The shuffle to Schiffmann algebra isomorphism. We use the same notation as in [3,4] for the Schiffmann algebra E of [5]. In our notation, E is generated by subalgebras Λ(X m,n ) isomorphic to the algebra Λ of symmetric functions over = Q(q, t), one for each pair of coprime integers m, n, and a central Laurent polynomial subalgebra [c ±1 1 , c ±1 2 ], subject to some defining relations.…”

Section: 2mentioning

confidence: 99%

LLT polynomials in the Schiffmann algebra

Blasiak¹,

Haiman²,

Morse³

et al. 2021

Preprint

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We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ(X m,n ) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra E of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr-Warrington conjecture in a companion paper to this one.

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“…The linear operator ∇ on Λ, introduced in [2], is defined to act diagonally in the basis of modified Macdonald polynomials Hµ (X; q, t) [11], with (5) ∇ Hµ = t n(µ) q n(µ * ) Hµ ,…”

Section: Catalanimals and Llt Polynomialsmentioning

confidence: 99%

“…For example, the head and foot at (3,5) in the den in Figure 1, shown as the upper left gray dot in Figure 2, prohibits the highest path π 4 from starting with an east step to (3,6), although nesting alone would allow this. Similarly, the head and foot at (5, 3) prohibits π 3 from passing through (5,4).…”

Section: Nests In a Den Formulamentioning

confidence: 99%

Dens, nests and the Loehr-Warrington conjecture

Blasiak¹,

Haiman²,

Morse³

et al. 2021

Preprint

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In a companion paper, we introduced raising operator series called Catalanimals. Among them are Schur Catalanimals, which represent Schur functions inside copies Λ(X m,n ) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra E of Burban and Schiffmann.Here we obtain a combinatorial formula for symmetric functions given by a class of Catalanimals that includes the Schur Catalanimals. Our formula is expressed as a weighted sum of LLT polynomials, with terms indexed by configurations of nested lattice paths called nests, having endpoints and bounding constraints controlled by data called a den.Applied to Schur Catalanimals for the alphabets X m,1 with n = 1, our 'nests in a den' formula proves the combinatorial formula conjectured by Loehr and Warrington for ∇ m s µ as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. When n is arbitrary, our formula establishes an (m, n) version of the Loehr-Warrington conjecture.In the case where each nest consists of a single lattice path, the nests in a den formula reduces to our previous shuffle theorem for paths under any line. Both this and the (m, n) Loehr-Warrington formula generalize the (km, kn) shuffle theorem proven by Carlsson and Mellit (for n = 1) and Mellit. Our formula here unifies these two generalizations.

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A proof of the Extended Delta Conjecture

Cited by 4 publications

References 19 publications

Springer fibers and the Delta Conjecture at $t=0$

Springer fibers and the Delta Conjecture at $t=0$

LLT polynomials in the Schiffmann algebra

Dens, nests and the Loehr-Warrington conjecture

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