Hessenberg variety

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.

Definitions

A Hessenberg function is a function of tuples

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

where

h(i+1) \geq \text{max }(i,h(i)) \text{ for all} 1 \leq i \leq n-1.

For example,

h(1,2,3,4,5)=(2,3,3,4,5) \,

is a Hessenberg function.

For any Hessenberg function h and a linear transformation

X: \C^n \rightarrow \C^n, \,

the Hessenberg variety is the set of all flags $F_{\bullet}$ such that

X \cdot F_i \subseteq F_{(h(i))}

for all i. Here $F_{(h(i))}$ denotes the vector space spanned by the first $h(i)$ vectors in the flag $F_{\bullet}$ .

\mathcal{H}(X,h) = \{ F_{\bullet} \mid X F_{i} \subset F_{(h_i)} \text{ for } 1 \leq i \leq n \}

Examples

Some examples of Hessenberg varieties (with their $h$ function) include:

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

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

The Springer variety: $h(i) = i$ for all $i$ .

References

F. De Mari, C. Procesi, and M. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (1992), 529–534.
B. Kostant, Flag Manifold Quantum Cohomology , the Toda Lattice, and the Representation with Highest Weight $\rho$ , Selecta Mathematica. (N.S.) 2, 1996, 43–91.
J. Tymoczko, Linear conditions imposed on flag varieties, Amer. J. Math. 128 (2006), 1587–1604.