Hessenberg varieties

Abstract:

Numerical algorithms involving Hessenberg matrices correspond to dynamical systems which evolve on the subvariety of complete flags ${S_1} \subset {S_2} \subset \cdots \subset {S_{n - 1}}$ in ${\mathbb {C}^n}$ satisfying the condition $s({S_i}) \subset {S_{i + 1}}$, $\forall i$, where $s$ is an endomorphism of ${\mathbb {C}^n}$. This paper describes the basic topological features of the generalization to subvarieties of $G/B$, $G$ a complex semisimple algebraic group, which are indexed by certain subsets of negative roots. In the special case where the subset consists of the negative simple roots, the variety coincides with the torus embedding associated to the decomposition into Weyl chambers.