A Hessenberg function is a map

${\displaystyle h:\{1,2,\ldots ,n\}\rightarrow \{1,2,\ldots ,n\}}$ 

such that

${\displaystyle h(i+1)\geq {\text{max }}(i,h(i))}$ 

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

${\displaystyle X:\mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n},\,}$ 

the Hessenberg variety ${\displaystyle {\mathcal {H}}(X,h)}$  is the set of all flags ${\displaystyle F_{\bullet }}$  such that

${\displaystyle X\cdot F_{i}\subseteq F_{h(i)}}$ 

for all i.

Some examples of Hessenberg varieties (with their ${\displaystyle h}$  function) include:

The Full Flag variety: h(i) = n for all i

The Peterson variety: ${\displaystyle h(i)=i+1}$  for ${\displaystyle i=1,2,\dots ,n-1}$ 

The Springer variety: ${\displaystyle h(i)=i}$  for all ${\displaystyle i}$ .

• De Mari, Filippo; Procesi, Claudio; Shayman, Mark A. (1992). "Hessenberg varieties". Transactions of the American Mathematical Society. 332 (2): 529–534. doi:10.1090/S0002-9947-1992-1043857-6. MR 1043857.
• Bertram Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ${\displaystyle \rho }$ , Selecta Mathematica (N.S.) 2, 1996, 43–91.
• Julianna Tymoczko, Linear conditions imposed on flag varieties, American Journal of Mathematics 128 (2006), 1587–1604.