We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Snmodules Rn,λ,s, which appear as the cohomology rings of the “∆Springer” varieties. These modules interpolate between the GarsiaProcesi 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 GarsiaProcesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing batterypowered 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.
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This content will become publicly available on May 10, 2025
Cocharge and Skewing Formulas for ΔSpringer Modules and the Delta Conjecture
We prove that $\omega \Delta ^{\prime}_{e_{k}}e_{n}_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a HallLittlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta $Springer modules. We use this to give an explicit Schur expansion in terms of the LascouxSchützenberger cocharge statistic on a new combinatorial object that we call a batterypowered tableau. Our proof is geometric, and shows that the $\Delta $Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^{2}$ coefficients of $\omega \Delta ^{\prime}_{e_{k}}e_{n}$.
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 Award ID(s):
 2054391
 NSFPAR ID:
 10524846
 Publisher / Repository:
 Oxford Academic
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Page Range / eLocation ID:
 rnae090
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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