Schedules and the Delta Conjecture
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Annals of Combinatorics
Schedules and the Delta Conjecture James Haglund and Emily Sergel Abstract. In a recent preprint, Carlsson and Oblomkov (Affine Schubert calculus and double coinvariants. arXiv preprint 1801.09033, 2018) obtain a long sought-after monomial basis for the ring DRn of diagonal coinvariants. Their basis is closely related to the “schedules” formula for the Hilbert series of DRn which was conjectured by the first author and Loehr (Discete Math 298(1–3):189–204, 2005) and first proved by Carlsson and Mellit (A proof of the shuffle conjecture. J Amer Math Soc 31(3):661–697, 2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article, we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018), which contains the Shuffle Theorem as a special case. Motivated by the Carlsson–Oblomkov basis for DRn and our Delta schedules formula, we introduce a (conjectural) basis for the super-diagonal coinvariant ring SDRn , an Sn -module generalizing DRn introduced recently by Zabrocki (a module for the Delta conjecture. arXiv preprint 1902.08966, 2019), which conjecturally corresponds to the Delta Conjecture. Mathematics Subject Classification. 05E10, 05E05. Keywords. Delta conjecture, Parking function, Coinvariant ring, Super-space.
1. Introduction Given a polynomial f (x1 , . . . , xn , y1 , . . . , yn ) ∈ C[x1 , . . . xn , y1 , . . . , yn ], the symmetric group Sn acts diagonally by permuting the x- and y-variables identically, i.e., for all σ ∈ Sn : σf (x1 , . . . , xn , y1 , . . . , yn ) = f (xσ1 , . . . , xσn , yσ1 , . . . , yσn ).
(1.1)
Let DRn denote the diagonal coinvariant ring, defined as the quotient: C[x1 , . . . xn , y1 , . . . , yn ]/In (X, Y ), 0123456789().: V,-vol
(1.2)
J. Haglund and E. Sergel
where In (X, Y ) is the ideal generated by all Sn -invariant polynomials in C[x1 , . . . xn , y1 , . . . , yn ] without constant term. A great deal of research over the last 25 years in algebra and combinatorics has been devoted to understanding the structure of DRn , starting with the original papers of Haiman [25] and Garsia-Haiman [14] introducing the topic. DRn is naturally bigraded by homogeneous x- and y-degrees, so (i,j) DRn = ⊕i,j≥0 DRn . A combinatorial description of the Hilbert series: Hilb(DRn ; q, t) = q i tj dim(DR(i,j) ) (1.3) n i,j≥0
in terms of parking functions was conjectured by the first author and Loehr [21]. The first author, Haiman, Loehr, Remmel, and Ulyanov [20], extended this to the famous Shuffle Conjecture (now the Shuffle Theorem), which takes into account the Sn action on DRn and gives a monomial expansion of the Frobenius characteristic Frob(DRn ; q, t). The Shuffle Theorem was proved only a few years ago by Carlsson and Mellit [7]. Here, the Frobenius characteristic is defined as: q i tj sλ (X) Mult(λ, DR(i,j) ), (1.4) Frob(DRn ; q, t) = n i,j≥0
λn
where the inner sum is over all partitions λ of n, sλ is the Schur function, and (i,j) Mult(λ
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