Affine pavings of Hessenberg varieties for semisimple groups

By Martha Precup. 2012

58 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/edc58927f476526b5b39bdf34a8751d38073ec41

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. T. Panov
6. D. Suh
7. Suyoung Choi
8. V. Buchstaber

In Geometry and Topology

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. T. Panov
6. D. Suh
7. Suyoung Choi

In Mathematical Physics

1. Hiraku Abe
2. Megumi Harada
3. M. Masuda
4. T. Horiguchi

Citations in the corpus

• Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “On the other hand, using her generalization [36] of Tymoczko’s computation of the Betti numbers of Hessenberg varieties in type A, Precup generalized our palindromicity results (Corollaries 36 and 37) to regular Hessenberg varieties for arbitrary complex semi-simple Lie algebras [35].”
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, background): “More generally, geometric properties and invariants of many different types of Hessenberg varieties (including in Lie types other than A) have been widely studied: for instance, the issue of connectedness is addressed in [52, 40], Tymoczko shows that they are not necessarily pure-dimensional in [52]”
• The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, unknown): “5 are combinatorial in nature, and rely on a formula for the Betti numbers of regular Hessenberg varieties obtained by the second author in [8] using geometric techniques.”