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Affine pavings of Hessenberg varieties for semisimple groups

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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.

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Correspondence to Martha Precup.

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Precup, M. Affine pavings of Hessenberg varieties for semisimple groups. Sel. Math. New Ser. 19, 903–922 (2013) doi:10.1007/s00029-012-0109-z

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Keywords

  • Hessenberg varieties
  • Affine paving
  • Bruhat decomposition

Mathematics Subject Classification (1991)

  • Primary 14L35
  • 14M15
  • Secondary 14F25