In geometry, Hessenberg varieties, first studied by De Mari, Procesi, and 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 Springer, Peterson, Kostant, among others, found connections with combinatorics, representation theory and cohomology.
Contents
Definitions
A Hessenberg function is a function of tuples
where
For example,
is a Hessenberg function.
For any Hessenberg function h and a linear transformation
the Hessenberg variety is the set of all flags
for all i. Here
Examples
Some examples of Hessenberg varieties (with their
The Full Flag variety: h(i) = n for all i
The Peterson variety:
The Springer variety: