Selecta Mathematica

, Volume 19, Issue 4, pp 903–922 | Cite as

Affine pavings of Hessenberg varieties for semisimple groups

Article

Abstract

In this paper, we consider certain closed subvarieties of the flag variety, known as Hessenberg varieties. We prove that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor are paved by affines. We provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements. As a consequence, we generalize results of Tymoczko asserting that Hessenberg varieties for regular nilpotent elements in the classical cases and arbitrary elements of \(\mathfrak{gl }_n(\mathbb C )\) are paved by affines. For example, our results prove that any Hessenberg variety corresponding to a regular element is paved by affines. As a corollary, in all these cases, the Hessenberg variety has no odd dimensional cohomology.

Keywords

Hessenberg varieties Affine paving Bruhat decomposition 

Mathematics Subject Classification (1991)

Primary 14L35 14M15 Secondary 14F25 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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