Cocharge and skewing formulas for $\Delta$-Springer modules and the Delta Conjecture

TL;DR

This paper proves new formulas connecting symmetric functions in the Delta Conjecture to skewing operators and Hall-Littlewood polynomials, introduces battery-powered tableaux, and provides geometric and combinatorial insights into $ ext{Springer}$ modules.

Contribution

It introduces a new combinatorial object called battery-powered tableaux and generalizes formulas for $ ext{Springer}$ modules using geometric methods.

Findings

Explicit Schur expansion in terms of cocharge statistic

Geometric proof linking $ ext{Springer}$ varieties to generalized Springer fibers

Conjectural formulas for coefficients in the Delta Conjecture

Abstract

We prove that $\omega \Delta'_{e_{k}}e_n|_{t=0}$, 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-Sch\"utzenberger cocharge statistic on a new combinatorial object that we call a \textit{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 $\omega \Delta'_{e_{k}}e_n$.