Decomposing Hessenberg varieties over classical groups

Abstract

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 a problem in numerical analysis and consists for a fixed linear operator M of the full flags V_1 \subsetneq V_2 >... \subsetneq V_n in GL_n with M V_i contained in V_{i+1} for all i. In this paper I show that all Hessenberg varieties in type A_n and semisimple and regular nilpotent Hessenberg varieties in types B_n,C_n, and D_n can be paved by affine spaces. Moreover, this paving is the intersection of a particular Bruhat decomposition with the Hessenberg variety. In type A_n, an equivalent description of the cells of the paving in terms of certain fillings of a Young diagram can be used to compute the Betti numbers of Hessenberg varieties. As an example, I show that the Poincare polynomial of the Peterson variety in A_n is \sum_{i =0}^{n-1} \binom{n-1}{i} x^{2i}.