Séminaire Lotharingien de Combinatoire, 89B.65 (2023), 12 pp.
Maria Gillespie, and Sean T. Griffin
A Cocharge Formula for the Δ-Springer Modules
Abstract.
We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Sn-modules Rn,λ,s, which appear as the cohomology rings of the "Δ-Springer" varieties. These modules interpolate between the Garsia-Procesi modules Rμ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture.
Our formula directly generalizes the known cocharge formula for Garsia-Procesi modules and gives a new cocharge formula for the Delta Conjecture at t=0, by introducing \textit{battery-powered tableaux} that ``store'' extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and
s ≤ ℓ(λ)+2, as well as for n ≤ 8 and
s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
The following versions are available: