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 $$ X: \Complex^n \rightarrow \Complex^n $$, 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 $$ F_{\bullet} $$. $$ \mathcal{H}(X,h) = \{ F_{\bullet} \mid X F_{i} \subset F_{(h_i)} \text{ for all } 1 \leq i \leq n \} $$

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.