We consider the problem of the compactification of the iso level sets of the Hessenberg matrices which is propounded by Ercolani et al. We determine the structure of the cohomology ring of the compact iso level set and obtain a new expression of the flag variety $G/B$.
"Compactifications of the iso level sets of the Hessenberg matrices and the full Kostant-Toda lattice." Proc. Japan Acad. Ser. A Math. Sci. 82 (7) 93 - 96, March 2006. https://doi.org/10.3792/pjaa.82.93