A proof of the theta operator conjecture

Abstract

In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Vanden Wyngaerd recently stated a conjecture relating two symmetric function operators, Dk and Θk. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the Dk operators. Even though we deal specifically with the relation between the Dk and Θk operators, our work introduces a method for finding relations between Θk and other plethystic operators which are important in this area of study.