Hessenberg variety

In geometry, Hessenberg varieties, first studied by Filippo De Mari, Claudio Procesi, and Mark A. Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg function "h" and a linear transformation "X". The study of Hessenberg varieties was first motivated by questions in numerical analysis in relation to algorithms for computing eigenvalues and eigenspaces of the linear operator "X". Later work by T. A. Springer, Dale Peterson, Bertram Kostant, among others, found connections with combinatorics, representation theory and cohomology.
Definitions.
A "Hessenberg function" is a map 
such that 
for each "i". For example, the function that sends the numbers 1 to 5 (in order) to 2, 3, 3, 4, and 5 is a Hessenberg function.
For any Hessenberg function "h" and a linear transformation 
the "Hessenberg variety" formula_4 is the set of all flags formula_5 such that 
for all "i".
Examples.
Some examples of Hessenberg varieties (with their formula_7 function) include:
The Full Flag variety: "h"("i") = "n" for all "i"
The Peterson variety: formula_8 for formula_9
The Springer variety: formula_10 for all formula_11.