New identities for Theta operators

A proof of the fermionic Theta coinvariant conjecture

“…We turn to the Theta operator side of Theorem 1.2. Our starting point is an identity of D'Adderio and Romero [5,Theorem 8.2]. We use the standard q-analogs…”

“…Building on the techniques in [3], D'Adderio and Mellit [4] used Theta operators to prove the rise version of the Delta Conjecture of Haglund, Remmel, and Wilson [8]. D'Adderio and Romero [5] rederived and clarified a slew of symmetric function identities using Theta operators, drastically shortening many of their proofs.…”

“…To obtain a recursive structure on the characters Frob (F DR n ) i,j , we prove a result (Lemma 3.3) on applying the skewing operators h ⊥ d to Kronecker products s λ * s µ which may be of independent interest for studying Kronecker products in general. For the right-hand side Θ i Θ j ∇e n−i−j | q=t=0 of Theorem 1.2, we apply a result of D'Adderio and Romero [5] for applying the operators h ⊥ d to expressions of the form Θ i Θ j H (n−i−j) (x; q, t) and study what happens at the evaluation q, t → 0. We check that the recursions coincide, and Theorem 1.2 is proven.…”