2021/09/01 by Sean T. Griffin, Griffin, Sean T., Jake Levinson +3 · 2 citations
Mathematics · #05E10 #05E14 #14F25 #14M15 (secondary) #20C30 (primary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Affine variety #Algebra over a field #Algebraic Geometry (math.AG) #Cohomology #Cohomology ring #Combinatorics #Combinatorics (math.CO) #Conjecture #Diagonal #Equivariant cohomology #FOS: Mathematics #Geometry #Intersection (aeronautics) #Lambda #Mathematics #Physics #Pure mathematics #Rank (graph theory) #Realization (probability) #Representation Theory (math.RT) #Ring (chemistry) #Variety (cybernetics)
paper · pdf · doi:10.48550/arxiv.2109.00639
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2021/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a family of varieties Yn,λ,s, which we call the Δ-Springer varieties, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H^*(Yn,λ,s) and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The λ=(1k) case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at t=0. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety Yn,λ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud--Saltman rank variety and diagonal matrices.