“Hessenberg variety”的意思、由来-开放百科全书

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Hessenberg variety

释义

Definitions
Examples
References

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

such that

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

the Hessenberg variety is the set of all flags such that

for all i.

Examples

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

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

The Peterson variety: for

The Springer variety: for all .

References

{{cite journal|first1=Filippo |last1=De Mari|first2=Claudio|last2=Procesi|authorlink2=Claudio Procesi | first3=Mark A. |last3=Shayman|title=Hessenberg varieties|journal=Transactions of the American Mathematical Society |volume=332|year=1992|issue=2|pages=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 , Selecta Mathematica (N.S.) 2, 1996, 43–91.
Julianna Tymoczko, Linear conditions imposed on flag varieties, American Journal of Mathematics 128 (2006), 1587–1604.

2 : Algebraic geometry|Algebraic combinatorics

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