Some combinatorial aspects of Theta and Delta operators
Marino Romero · Contemporary mathematics - American Mathematical Society · 2025
Theta operators are essential in the statement and proof of the Compositional Delta Conjecture, and they are also used in describing the conjectural Frobenius characteristic for the S n S_n -coinvariants of the polynomial ring with two commuting and two anticommuting sets of variables. We will look at how two of the most recent advancements in the theory of Macdonald polynomials, and related symmetric function operators, describe Theta operators. The first is in regards to the proof of the Delta Conjecture, by D’Adderio and Mellit, and the five-term relation of Garsia and Mellit. The second is with regards to the recent advances by Blasiak, Haiman, Morse, Pun, and Seelinger, regarding connections between the Schiffmann Algebra, the Shuffle Algebra, and LLT polynomials. We end by describing new results regarding Theta and Delta operators, and we present some open questions whose answers may lay hidden between the connections of these advancements.