Decomposing Nilpotent Hessenberg Varieties over Classical Groups

Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from numerical analysis and consists, for a fixed linear operator M , of the full flags V1 ( V2 . . . ( Vn in GLn with MVi ⊆ Vi+1 for all i. In this paper, I show that all nilpotent Hessenberg varieties in type A and regular nilpotent Hessenberg varieties in types B,C, and D can be paved by affines. Moreover, this paving is induced by a particular Bruhat decomposition on the Hessenberg variety. In type A, an equivalent combinatorial description of the cells of the paving in terms of certain fillings of a Young diagram can be used to determine the dimension of the cell. As an example, I show that the Poincare polynomial of the Peterson variety in An is ∑ n−1 i=0 (