• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Topology of Hessenberg varieties and representations of symmetric groups

Research Project

Project/Area Number 15K17544
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Geometry
Research InstitutionOsaka City University

Principal Investigator

Abe Hiraku  大阪市立大学, 大学院理学研究科, 数学研究所専任研究所員 (00736499)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywords旗多様体 / Hessenberg多様体 / トーリック多様体 / コホモロジー環 / 対称群の表現 / ポアンカレ双対代数 / regular semisimple / regular nilpotent
Outline of Final Research Achievements

We studied the ring structures of the cohomology of Hessenberg varieties.
(1)We explicitly determined the cohomology ring of an arbitrary regular nilpotent Hessenberg variety in Lie type A, and we also proved that it is isomorphic (as rings) to the symmetric group invariant subring of the cohomology ring of the corresponding regular semisimple Hessenberg variety. (2)For a Hessenberg function of special form, we explicitly determined the cohomology ring of the regular semisimple Hessenberg variety in Lie type A. (3)We gave a certain presentation of the cohomology ring of an arbitrary Hessenberg variety for the minimal nilpotent orbit.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (23 results)

All 2018 2017 2016 2015 Other

All Int'l Joint Research (3 results) Journal Article (3 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 3 results,  Acknowledgement Compliant: 1 results) Presentation (16 results) (of which Int'l Joint Research: 14 results,  Invited: 9 results) Funded Workshop (1 results)

  • [Int'l Joint Research] McMaster University(Canada)

    • Related Report
      2017 Annual Research Report
  • [Int'l Joint Research] McMaster University(Canada)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] University of Toronto(Canada)

    • Related Report
      2016 Research-status Report
  • [Journal Article] The cohomology rings of regular semisimple Hessenberg varieties for h=(h(1),n,...,n)2018

    • Author(s)
      Hiraku Abe, Tatsuya Horiguchi, Mikiya Masuda
    • Journal Title

      Journal of Combinatorics

      Volume: 印刷中

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] THE COHOMOLOGY RINGS OF REGULAR NILPOTENT HESSENBERG VARIETIES IN LIE TYPE A2018

    • Author(s)
      Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda
    • Journal Title

      International Mathematics Research Notices

      Volume: 16 Issue: 05 Pages: 1-52

    • DOI

      10.1093/imrn/rnx275

    • URL

      https://ocu-omu.repo.nii.ac.jp/records/2016845

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Hessenberg varieties for the minimal nilpotent orbit2017

    • Author(s)
      Hiraku Abe and Peter Crooks
    • Journal Title

      Pure Appl. Math. Q.

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] 正則半単純なヘッセンバーグ多様体のコホモロジー環について2018

    • Author(s)
      阿部拓
    • Organizer
      日本数学会2018年度年会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hessenberg varieties I2017

    • Author(s)
      Hiraku Abe
    • Organizer
      Toric topology in Osaka 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Cohomology of Hessenberg varieties and representations of symmetric groups2017

    • Author(s)
      Hiraku Abe
    • Organizer
      International Festival in Schubert Calculus
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Extensions of pavings by affines of Peterson varieties2017

    • Author(s)
      Hiraku Abe
    • Organizer
      Young Researchers in Homotopy Theory and Topic Topology
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Flat families of Hessenberg varieties with an application to Newton-Okounkov bodies2017

    • Author(s)
      Hiraku Abe
    • Organizer
      AMS special session
    • Place of Presentation
      Atlanta (United States of America)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Flat families of Hessenberg varieties with an application to Newton-Okounkov bodies2017

    • Author(s)
      Hiraku Abe
    • Organizer
      Geometry and Topology seminar
    • Place of Presentation
      The University of Western Ontario (Canada)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The cohomology of Hessenberg varieties and representations of symmetric groups2016

    • Author(s)
      Hiraku Abe
    • Organizer
      Geometry & Topology seminar
    • Place of Presentation
      Hamilton (Canada)
    • Year and Date
      2016-02-25
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the flatness of certain families of Hessenberg varieties2016

    • Author(s)
      Hiraku Abe
    • Organizer
      Toric Topology 2016 in Kagoshima
    • Place of Presentation
      鹿児島大学(鹿児島県鹿児島市郡元)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On flat degenerations of regular semisimple Hessenberg varieties2016

    • Author(s)
      Hiraku Abe
    • Organizer
      Mini-Workshop on Toric Topology in Okayama
    • Place of Presentation
      岡山理科大学(岡山県岡山市北区理大町)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Flat families of Hessenberg varieties with an application to Newton-Okounkov bodies2016

    • Author(s)
      Hiraku Abe
    • Organizer
      algebra/combinatorics/geometry research seminar
    • Place of Presentation
      Smith college (United States of America)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The cohomology of regular Hessenberg varieties and representations of symmetric groups2015

    • Author(s)
      Hiraku Abe
    • Organizer
      A special session on Algebraic Geometry and Combinatorics at the AMS Sectional Meeting
    • Place of Presentation
      New Brunswick (United States of America)
    • Year and Date
      2015-11-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Nilpotent vs. semisimple via representations of symmetric groups2015

    • Author(s)
      Hiraku Abe
    • Organizer
      Workshop on Recent Developments in the Geometry and Combinatorics of Hessenberg Varieties
    • Place of Presentation
      Toronto (Canada)
    • Year and Date
      2015-07-23
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] Representations of symmetric groups on the cohomology of Hessenberg varieties2015

    • Author(s)
      Hiraku Abe
    • Organizer
      Toric Topology 2015 in Osaka
    • Place of Presentation
      大阪市立大学(大阪市住吉区杉本)
    • Year and Date
      2015-06-18
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] ヘッセンバーグ多様体のコホモロジー環と対称群の表現2015

    • Author(s)
      阿部拓
    • Organizer
      幾何学・組合せ論に現れる環と代数構造
    • Place of Presentation
      京都大学数理解析研究所(京都市左京区北白川追分町)
    • Year and Date
      2015-06-11
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] The cohomology of Hessenberg varieties and representations of symmetric group2015

    • Author(s)
      Hiraku Abe
    • Organizer
      Symplectic Geometry Seminar
    • Place of Presentation
      Toronto (Canada)
    • Year and Date
      2015-06-01
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] The cohomology of Hessenberg varieties and representations of symmetric groups2015

    • Author(s)
      Hiraku Abe
    • Organizer
      Topology and Geometric Group Theory Seminar
    • Place of Presentation
      Ithaca (United States of America)
    • Year and Date
      2015-04-28
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Funded Workshop] Toric topology in Osaka 20172017

    • Related Report
      2017 Annual Research Report

URL: 

Published: 2015-04-16   Modified: 2019-03-29  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi