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Action of groups to tensors and their ring of invariants

Research Project

Project/Area Number 15K04818
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionKyoto University of Education

Principal Investigator

Mitsuhiro Miyazaki  京都教育大学, 教育学部, 准教授 (90219767)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywords高次元配列 / テンソル / 群作用 / 不変式 / サグビー基底 / 日比環 / グレブナー基底
Outline of Final Research Achievements

High dimensional array of data are called tensors in the field of data analysis. We considered in this research group actions to tensors with entries in a commutative ring.
We found that in many situations, there are rings of invariants whose initial algebra is the Ehrhart ring of a convex polytope. In particular, there are many cases that such Ehrhart rings are Hibi rings. Thus, we studied Hibi rings and obtained several results. Further, we defined a notion Doset Hibi ring and showed that the initial algebra of a certain rings of invariants have structures of Doset Hibi rings.

Report

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

    (25 results)

All 2018 2017 2016 2015

All Journal Article (10 results) (of which Peer Reviewed: 7 results,  Acknowledgement Compliant: 2 results,  Open Access: 1 results) Presentation (14 results) (of which Int'l Joint Research: 8 results,  Invited: 3 results) Book (1 results)

  • [Journal Article] Almost Gorenstein Hibi rings2018

    • Author(s)
      Mitsuhiro Miyazaki
    • Journal Title

      Journal of Algebra

      Volume: 493 Pages: 135-149

    • DOI

      10.1016/j.jalgebra.2017.09.033

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals2017

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Journal Title

      Journal of Algebra

      Volume: 471 Pages: 409-453

    • DOI

      10.1016/j.jalgebra.2016.09.028

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Action of special linear groups to the tensor of indeterminates and classical invariants of binary forms2017

    • Author(s)
      Mitsuhiro Miyazaki
    • Journal Title

      Journal of Algebra

      Volume: 473 Pages: 430-446

    • DOI

      10.1016/j.jalgebra.2016.11.021

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the generators of the canonical module of a Hibi ring: a criterion of level property and the degrees of generators2017

    • Author(s)
      Mitsuhiro Miyazaki
    • Journal Title

      Journal of Algebra

      Volume: 480 Pages: 215-236

    • DOI

      10.1016/j.jalgebra.2017.02.011

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Typical ranks of certain 3-tensors and absolutely full column rank tensors2017

    • Author(s)
      Mitsuhiro Miyazaki, Toshio sumi and Toshio Sakata
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 印刷中 Issue: 1 Pages: 193-205

    • DOI

      10.1080/03081087.2017.1292994

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Doset Hibi rings with an application to invariant theory2017

    • Author(s)
      Mitsuhiro Miyazaki
    • Journal Title

      Communications in Algebra

      Volume: 印刷中 Issue: 1 Pages: 335-355

    • DOI

      10.1080/00927872.2017.1321654

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Determinantal varieties, Bezout's theorem and typical ranks of 3-tensors over the real number filed2017

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata
    • Journal Title

      第38回可換環論シンポジウム報告集

      Volume: 38 Pages: 197-203

    • Related Report
      2016 Research-status Report
  • [Journal Article] On the real radical of certain determinantal ideals defined by tensors2016

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata
    • Journal Title

      第37回可換環論シンポジウム報告集

      Volume: 37 Pages: 121-131

    • Related Report
      2015 Research-status Report
  • [Journal Article] Typical rank of mxnx(m-1)n tensors with 3<=m<=n over the real number field2015

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Journal Title

      Linear and Multilinear Algebra

      Volume: 63(5) Issue: 5 Pages: 940-955

    • DOI

      10.1080/03081087.2014.910206

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Tensor of indeterminates への群作用とその不変式環のinitial algebra について2015

    • Author(s)
      宮崎充弘
    • Journal Title

      第27回可換環論セミナー報告集

      Volume: 27 Pages: 103-107

    • Related Report
      2015 Research-status Report
  • [Presentation] On the anticanonical level property of a Hibi ring2018

    • Author(s)
      宮崎 充弘
    • Organizer
      日本数学会2018年度年会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hibi ringのcanonical idealのanalytic spreadとanti-canonical symbolic spreadについて2018

    • Author(s)
      宮崎充弘
    • Organizer
      Commutative Algebra Day in Kyoto
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hibi ringのcanonical idealに関するRees algebraとanticanonical coverについて2018

    • Author(s)
      宮崎 充弘
    • Organizer
      第30回可換環論セミナー
    • Related Report
      2017 Annual Research Report
  • [Presentation] Hibi rings2018

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      International Workshop on Commutative Algebra
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Negative and positive symbolic powers of the canonical ideal of a Hibi ring2017

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      第39回可換環論シンポジウム
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] On the almost Gorenstein property of Hibi rings2017

    • Author(s)
      宮崎 充弘
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2017 Annual Research Report
  • [Presentation] Actions of special linear groups to tensors of indeterminates and standard property of a certain Ehrhart ring2017

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      日本数学会年会
    • Place of Presentation
      首都大学東京
    • Related Report
      2016 Research-status Report
  • [Presentation] On the real radical of certain determinantal ideals and multiple typical ranks of 3-tensors over the real number field2016

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      International Conference and 8th Japan-Vietnam joint Seminar on Commutative Algebra
    • Place of Presentation
      Ha-Long, Vietnam
    • Year and Date
      2016-03-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] ある種のテンソルにより定義される行列の行列式イデアルの実根基について2016

    • Author(s)
      宮崎充弘
    • Organizer
      日本数学会年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] Determinantal varieties and typical ranks of 3-tensors2016

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Organizer
      Applications of Algebraic Methods to Statistics
    • Place of Presentation
      Research Institute of Mathematical Science, Kyoto University
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Typical Ranks of Tensors Over the Real Number Field and Determinantal Ideals2016

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Organizer
      The 4th Institute of Mathematical Statistics Asia Pacific Rim Meeting
    • Place of Presentation
      The Chinese University of Hong Kong, Hong Kong
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] Determinantal varieties, Bezout's Theorem and typical ranks of 3-tensors over the real number field2016

    • Author(s)
      Toshio Sumi, Mitsuhiro Miyazaki and Toshio Sakata
    • Organizer
      第38回可換環論シンポジウム
    • Place of Presentation
      IPC 生産性国際交流センター(神奈川県三浦郡葉山町湘南国際村)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] On the real radical of certain determinantal ideals defined by tensors2015

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      第37回可換環論シンポジウム
    • Place of Presentation
      倉敷シーサイドホテル(岡山県倉敷市)
    • Year and Date
      2015-11-21
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] About typical ranks of real tensors2015

    • Author(s)
      Mitsuhiro Miyazaki
    • Organizer
      SIAM Conference on Applied Algebraic Geometry
    • Place of Presentation
      National Institute for Mathematical Sciences, 韓国、Daejeon(大田)
    • Year and Date
      2015-08-04
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Book] Algebraic and Computational Aspects of Real Tensor Ranks2016

    • Author(s)
      Toshio Sakata, Toshio Sumi and Mitsuhiro Miyazaki
    • Total Pages
      116
    • Publisher
      Springer
    • Related Report
      2015 Research-status Report

URL: 

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

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