The Delta Conjecture

  title={The Delta Conjecture},
  author={James Haglund and Jeffrey B. Remmel and Andrew Timothy Wilson},
  journal={Discrete Mathematics \& Theoretical Computer Science},
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… 

A proof of the Extended Delta Conjecture

We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for ∆hl∆ ′ ek en, where ∆ ′ ek and ∆hl are Macdonald eigenoperators and en is an elementary symmetric


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

Some consequences of the valley Delta conjectures

. In [16] Haglund, Remmel and Wilson introduced their Delta conjectures , which give two different combinatorial interpretations of the symmetric function ∆ ′ e n − k − 1 e n in terms of

Ordered set partition statistics and the Delta Conjecture

Combinatorial Aspects of Macdonald and Related Polynomials

The theory of symmetric functions plays an increasingly important role in modern mathematics, with substantial applications to representation theory, algebraic geometry, special functions,

A minimaj-preserving crystal on ordered multiset partitions

Decorated Dyck paths and the Delta conjecture

. We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework in (Haglund et al 2015) and (Zabrocki 2016), by introducing two new statistics, bounce and

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

We introduce a family of varieties Yn,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(Yn,λ,s) and

Schedules and the Delta Conjecture

In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the



Generalized Shuffle Conjectures for the Garsia-Haiman Delta Operator

We conjecture two combinatorial interpretations for the symmetric function Delta/ek en, where [Delta]f is an eigenoperator for the modified Macdonald polynomials defined by Garsia and Haiman. The

A combinatorial formula for the character of the diagonal coinvariants

Author(s): Haglund, J; Haiman, M; Loehr, N; Remmel, J B; Ulyanov, A | Abstract: Let Rn be the ring of coinvariants for the diagonal action of the symmetric group Sn. It is known that the character of

A proof of the q,t-Schröder conjecture

We prove a recent conjecture of Egge, Haglund, Killpatrick and Kremer (Elec. J. Combin. 10 (2003), #R16), which gives a combinatorial formula for the coefficients of a hook shape in the Schur

A weighted sum over generalized Tesler matrices

We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these

A proof of the shuffle conjecture

We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial

Ordered set partition statistics and the Delta Conjecture

A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Abstract We introduce a $q,\,t$ -enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action

An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions

This work proves a strengthening of the theorem that inversion number and major index have the same distribution over permutations of a given multiset, and uses the main theorem to show that these polynomials are symmetric and the Schur expansion.

Compositional (km,kn)-Shuffle Conjectures

In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of