Cocharge and skewing formulas for Δ-Springer modules and the Delta Conjecture

2023/07/05 by Maria Gillespie, Sean T. Griffin, Gillespie, Maria +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2307.02645

openalex publication_date 2023/07/05 · openalex created_date 2023/08/01 · openalex updated_date 2026/07/28

Abstract

We prove that ωΔ'_e_ke_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 Δ-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 Δ-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 t² coefficients of ωΔ'_e_ke_n.

Cited by

• Representation Theoretic Bases for the Δ-Springer Module

Related

• Δ-Springer varieties and Hall-Littlewood polynomials
• Springer fibers and the Delta Conjecture at t=0
• A geometric interpretation of the Delta Conjecture
• A geometric interpretation for the Delta Conjecture
• Exploring a Delta Schur Conjecture