The Delta Conjecture

Abstract

International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

A proof of the Extended Delta Conjecture

“…Remark 2.2.3. In [14], a Dyck path is a north-east lattice path lying weakly above the line segment connecting (0, 0) and (N, N), and labellings increase from south to north along vertical runs. After reflecting the picture about a horizontal line, our conventions on paths, labellings, and the definition of dinv(P ) match those in [14].…”

“…In [14], a Dyck path is a north-east lattice path lying weakly above the line segment connecting (0, 0) and (N, N), and labellings increase from south to north along vertical runs. After reflecting the picture about a horizontal line, our conventions on paths, labellings, and the definition of dinv(P ) match those in [14]. Separately, [13] uses the same conventions that we do for Dyck paths, but defines labellings to increase from south to north, and defines dinv(P ) with the inequalities in (9) reversed.…”

“…We prove the Extended Delta Conjecture of Haglund, Remmel and Wilson [14] by adapting methods from our work in [2] on a generalized Shuffle Theorem and proving new results about the action of the elliptic Hall algebra on symmetric functions. As in [2], we reformulate the conjecture as the polynomial truncation of an identity of infinite series of GL m characters, expressed in terms of LLT series.…”

“…An expanded investigation led Haglund, Remmel and Wilson [14] to the Delta Conjecture, a combinatorial prediction for ∆ ′ e k e n , for all 0 ≤ k < n. This led to a flurry of activity (e.g. [6,11,14,15,16,20,21,22,25,27]), including a conjecture by Zabrocki [26] that ∆ ′ e k e n captures the character of the super-diagonal coinvariant ring SDR n , a deformation of DR n involving the addition of a set of anti-commuting variables.…”

“…Conjecture [14,Conjecture 7.4] is, for l ≥ 0 and 1 ≤ k ≤ n, (1) ∆ h l ∆ ′ e k−1 e n = z n−k λ∈D n+l P ∈L n+l,l (λ)…”