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

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

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.

Examples

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 ${\displaystyle i}$.

References

• 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.