Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies

Abstract

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and the special fiber is a regular nilpotent Hessenberg variety, are flat and have reduced fibres. This result further allows us to give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Furthermore, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, which are suitable for computing Newton-Okounkov bodies. As an application of our results, we explicitly compute many Newton-Okounkov bodies of the two-dimensional Peterson variety with respect to Plücker embeddings.

25 pages. Corrected minor errors in Example 3.5 and in Sections 6 and 7. The main results remain the same

References in corpus (2)

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• Hessenberg varieties and hyperplane arrangements

Cited by in corpus (10)

• Hessenberg varieties and hyperplane arrangements
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$
• Geometry of regular Hessenberg varieties
• A survey of recent developments on Hessenberg varieties
• Geometric vertex decomposition, Gröbner bases, and Frobenius splittings for regular nilpotent Hessenberg varieties
• Gröbner geometry for regular nilpotent Hessenberg Schubert cells
• Hessenberg varieties, Slodowy slices, and integrable systems
• Singular string polytopes and functorial resolutions from Newton-Okounkov bodies
• The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope
• Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell