Hessenberg variety
Introduction 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''  is the set of all flags  such that : for all ''i''.
Examples Some examples of Hessenberg varieties (with their  function) include:The Full Flag variety: ''h''(''i'') = ''n'' for all ''i''The Peterson variety:  for The Springer variety:  for all .
References  **Bertram Kostant, ''Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ,'' Selecta Mathematica (N.S.) '''2''', 1996, 43–91. *Julianna Tymoczko, ''Linear conditions imposed on flag varieties'', American Journal of Mathematics '''128''' (2006), 1587–1604.