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 “∆-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 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.
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This content will become publicly available on May 10, 2025
Cocharge and Skewing Formulas for Δ-Springer 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 Hall-Littlewood 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 Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered 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
- NSF-PAR ID:
- 10524846
- Publisher / Repository:
- Oxford Academic
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- rnae090
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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