The Delta Conjecture
By James Haglund, Jeffrey B. Remmel, Andrew Timothy Wilson. Discrete Mathematics & Theoretical Computer Science, 2015
121 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/1bd801ec08ae0da44cfa19368cba755956420baf
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Citations in the corpus
- A module for the Delta conjecture. (2019, background): “ension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in ‘the Delta conjecture’ of Haglund, Remmel and Wilson [HRW]. The following account of the history of the conjecture stated here is necessarily abbreviated”