Representation Theoretic Bases for the Δ-Springer Module

2025/09/29 by Chou, Raymond, Mitsuki Hanada, Hanada, Mitsuki
Mathematics · Physics and Astronomy · #05A30 #05E10 #20C30 #Algebraic and Geometric Analysis #Combinatorics (math.CO) #FOS: Mathematics #Quantum Mechanics and Applications #Representation Theory (math.RT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2509.24252

openalex publication_date 2025/09/29 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We give a descent monomial basis of Δ-Springer modules R_n,λ,s, first defined by Griffin. Our construction simultaneously generalizes the descent basis for the Garsia-Procesi module R_λ studied by Carlsson-Chou and Hanada, as well as the descent basis for the generalized coinvariant algebras R_n,k studied by Haglund-Rhoades-Shimozono. This basis is deeply connected with a combinatorial object called battery-powered tableaux, introduced by Gillespie-Griffin. We highlight the representation theoretic properties of this monomial basis by using it to give a direct combinatorial proof of the graded Frobenius character of R_n,λ,s in terms of battery-powered tableaux, a fact which has only the geometric proof of Gillespie-Griffin. We also conjecture a higher Specht basis of R_n,λ,s, generalizing the higher Specht basis of the coinvariant ring defined in Ariki-Terasoma-Yamada. This construction coincides with the Gillespie-Rhoades higher Specht basis for R_n,k. We give a proof for when λ= (λ₁,λ₂) is a partition of two rows.

Citations

• A combinatorial skewing formula for the Rise Delta Theorem
• A descent basis for the Garsia-Procesi module
• Cocharge and skewing formulas for Δ-Springer modules and the Delta Conjecture
• Springer fibers and the Delta Conjecture at t=0
• Higher Specht bases for generalizations of the coinvariant ring
• Harmonic bases for generalized coinvariant algebras
• A flag variety for the Delta Conjecture
• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
• An insertion algorithm for catabolizability
• Descent Representations and Multivariate Statistics
• Graded characters of modules supported in the closure of a nilpotent conjugacy class

Related

• Descent Representations of Generalized Coinvariant Algebras
• A descent basis for the Garsia-Procesi module
• Higher Specht bases for generalizations of the coinvariant ring
• A charge monomial basis of the Garsia-Procesi ring
• Constructing a Gröbner basis of Griffin's ideal