Hessenberg varieties for the minimal nilpotent orbit

TL;DR

This paper studies Hessenberg varieties linked to the minimal nilpotent orbit in complex simple algebraic groups, computing their topological invariants and describing their cohomology rings, especially in Lie type A.

Contribution

It provides explicit calculations of Poincaré polynomials, irreducible components, GKM graph structures, and cohomology rings for these Hessenberg varieties, advancing understanding in Lie theory and algebraic geometry.

Findings

Computed Poincaré polynomials for Hessenberg varieties in Lie type A.

Determined irreducible components of these varieties.

Explicitly described GKM graphs and cohomology rings.

Abstract

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 Poincar\'{e} 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.