2026/07/28 by Zijing Zhuang
Mathematics · #math.AG
We study the type A nilpotent Hessenberg varieties associated with Hessenberg functions that satisfy h(i)≤ i+1. We call these the generalized parabolic Peterson varieties. We show that such varieties can be decomposed into the union of specific generalized parabolic Peterson varieties Petλ,α, such that λ is an integer partition, α is an integer composition, and α is dominated by λ. We prove that when α is dominated by λ, the cardinality of the maximal dimensional components of Petλ,α equals the Kostka number Kλα, and its dimension is determined only by λ and the length of α. We provide a recursive formula for the Poincaré polynomial of Petλ,α. These results partially answer open questions about the geometry of Hessenberg varieties but raise further questions about the representation-theoretic reasons for these facts.