User:Fkdtrtl03/Hessenberg variety

First studied by De Mari, Procesi, and Shayman, Hessenberg varieties 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 in an effort to find efficient algorithms for computing eigenvalues and eigenspaces of the linear operator $X$. Later T. Springer studied Hessenberg varieties and their connections with representations of the Weyl group. Kostant showed that the quantum cohomology of the flag variety coincides with a coordinate ring of a particular subvariety of the Peterson variety.

Definitions

A Hessenberg function is a function of tuples $h:\{1,2,\ldots ,n\}\rightarrow \{1,2,\ldots ,n\}$ where $h(i)\leq h(i+1)$ 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 $X:\mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n}$, 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 $F_{\bullet }$.

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

Examples

For the function $h(1,2,\ldots ,n)=(1,2,\ldots ,n)$ the variety ${\mathcal {H}}(X,h)$ associated with $h$ is commonly called the Springer variety.

For $h(1,2,\ldots n)=(n,n,\ldots ,n)$ the Hessenberg variety is the full flag variety ${\mathcal {H}}(X,h)$.

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.