Hessenberg Varieties Research Articles

Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix

Let A∈gl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S 1 ⊂⋯⊂S n−1 in ℂn which satisfy the condition AS i ⊂S i+p ,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, |λ1|<⋯<|λ1|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A 0=P -1 AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A 0 converge to a given permutation of λ1,λn.

Hessenberg varieties for the minimal nilpotent orbit

For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincare polynomials and irreducible components of these varieties in Lie type $A$. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus $T\subseteq G$. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and $T$-equivariant cohomology rings of our varieties as quotients of those of the flag variety.