Polarization Operators in Superspace

The classical coinvariant rings and its variants are quotient rings with rich connections to combinatorics, symmetric function theory and geometry. Studies of a generalization of the classical coinvariant rings known as the diagonal harmonics have fruitfully produced many interesting discoveries in combinatorics including the q, t-Catalan numbers and the Shuffle Theorem.

The super coinvariant rings are a direct generalization of the classical coinvariant rings to one set of commuting variables and one set of anticommuting variables. N. Bergeron, Li, Machacek, Sulzgruber, and Zabrocki conjectured in 2018 that the super coinvariant rings are representation theoretic models for the Delta Conjecture at t = 0.

In this dissertation, we explore the super coinvariant rings using algebraic and combina- torial methods. In particular, we study the alternating component of the super harmonics and discover a novel basis using polarization operators. We use polarization equivalence to establish a triangularity relation between the new basis and a known basis due to two groups of researchers Bergeron, Li, Machacek, Sulzgruber, and Zabrocki and Swanson and Wallach. Furthermore, we prove a folklore result on the cocharge statistics of standard Young tableaux and propose a basis for every irreducible representation appearing in the super harmonics.