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 to 2, 3, 3, 4, and 5 is a Hessenberg function.
For any Hessenberg function h and a linear transformation
the Hessenberg variety is the set of all flags such that
for all i.

Examples

Some examples of Hessenberg varieties include:
The Full Flag variety: h = n for all i
The Peterson variety: for
The Springer variety: for all.