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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

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Abstract

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S4K1, Canada

    Hiraku Abe, Federico Galetto & Megumi Harada

  2. Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Hiraku Abe

  3. Department of Mathematics and Statistics, University of Calgary, 2500 University Dr. NW, Calgary, AB, T2N1N4, Canada

    Lauren DeDieu

Authors
  1. Hiraku Abe
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  2. Lauren DeDieu
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  3. Federico Galetto
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  4. Megumi Harada
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Correspondence to Hiraku Abe.

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Abe, H., DeDieu, L., Galetto, F. et al. Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies. Sel. Math. New Ser. 24, 2129–2163 (2018). https://doi.org/10.1007/s00029-018-0405-3

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