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McKay correspondence and its expansion from the viewpoint of cluster theory

Research Project

Project/Area Number 20K14279
Research Category

Grant-in-Aid for Early-Career Scientists

Allocation TypeMulti-year Fund
Review Section Basic Section 11010:Algebra-related
Research InstitutionKyoto Sangyo University

Principal Investigator

Nakajima Yusuke  京都産業大学, 理学部, 助教 (20783096)

Project Period (FY) 2020-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2023: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywordsクレパント特異点解消 / 非可換クレパント特異点解消 / マッカイ対応 / ダイマー模型 / 団理論・団傾理論 / 変異 / トーリック特異点 / 安定性条件 / cDV特異点 / 箙の表現 / modifying加群 / 団理論 / 組合せ的変異 / グラスマン多様体 / トーリック退化 / トーリック多様体 / 非可換特異点解消 / 極大Cohen-Macaulay加群
Outline of Research at the Start

マッカイ対応は, 2次元商特異点を介して, 異なる数学的対象が結びつく現象であり, 現在では, 様々な特異点に対する類似・拡張が研究されている.この対応を拡張しようとする際には, 団傾加群という対象が現れる. また, 団傾加群の変異という操作から導かれる「団構造」が, 多くの数学分野に現れることが近年明らかになっている. 本研究では, 団傾加群と, その変異を理解することを通じて「マッカイ対応の拡張」および「特異点を介した数学分野の新たな結びつきの発見」を目指す.

Outline of Final Research Achievements

When we extend the classical McKay correspondence, some representation theoretic objects such as non-commutative crepant resolutions and cluster tilting modules appear. For these objects, we can define an operation called "mutation", which reveals connections with the theory of cluster algebras. In this study, we focused on mutations of some objects associated with toric singularities/varieties, and observed their underlying cluster structures and the relationships with the McKay correspondence.
The main results are as follows: (1) Establishment of the operation "deformation of dimer models" that induces the combinatorial mutation of polygons (2) Investigation of the relationship between toric degenerations of Grassmannians and combinatorial mutations (3) Clarification of stability conditions that give crepant resolutions of toric cDV singularities and their connection to zigzag paths on a dimer model.

Academic Significance and Societal Importance of the Research Achievements

本研究のトピックのひとつである「マッカイ対応」の発見以来、マッカイ対応を通じて様々な分野の関係性が見出されてきた。その関係性は新たな研究視点をもたらし、代数幾何・環論・表現論などの分野の発展につながっている。本研究で注目した「団理論」も同様に、複数の分野の背後に隠れている共通の構造を見出すものであり、分野の垣根を超えた研究が進んでいる。
本研究においては、マッカイ対応、団理論に関わる新たな研究成果を上げており、その成果は関連分野の研究に大きく寄与すると考える。また、本研究に現れるダイマー模型は、超弦理論・ミラー対称性といった物理学に関わる話題との親和性も高く、研究のさらなる広がりが見込まれる。

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (24 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (2 results) Journal Article (6 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 5 results,  Open Access: 3 results) Presentation (15 results) (of which Int'l Joint Research: 5 results,  Invited: 12 results) Remarks (1 results)

  • [Int'l Joint Research] University of Nebraska, Lincoln(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Basque Center for Applied Mathematics(スペイン)

    • Related Report
      2022 Research-status Report
  • [Journal Article] Wall-and-chamber structures of stability parameters for some dimer quivers2024

    • Author(s)
      Yusuke Nakajima
    • Journal Title

      The proceedings of the 55th Symposium on Ring Theory and Representation Theory

      Volume: - Pages: 58-65

    • Related Report
      2023 Annual Research Report
    • Open Access
  • [Journal Article] Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs2023

    • Author(s)
      Akihiro Higashitani, Yusuke Nakajima
    • Journal Title

      Advanced Studies in Pure Mathematics, McKay Correspondence, Mutation and Related Topics

      Volume: 88 Pages: 227-278

    • DOI

      10.2969/aspm/08810227

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Deformations of Dimer Models2022

    • Author(s)
      Higashitani Akihiro、Osaka University, Japan、Nakajima Yusuke、Kyoto Sangyo University, Japan
    • Journal Title

      Symmetry, Integrability and Geometry: Methods and Applications

      Volume: 18

    • DOI

      10.3842/sigma.2022.030

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On 2-Representation Infinite Algebras Arising From Dimer Models2022

    • Author(s)
      Yusuke Nakajima
    • Journal Title

      The Quarterly Journal of Mathematics

      Volume: 73 Issue: 4 Pages: 1517-1553

    • DOI

      10.1093/qmath/haac016

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Lower bounds on Hilbert-Kunz multiplicities and maximal F-signatures2022

    • Author(s)
      JACK JEFFRIES, YUSUKE NAKAJIMA, ILYA SMIRNOV, KEI-ICHI WATANABE , KEN-ICHI YOSHIDA
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society

      Volume: 174 Issue: 2 Pages: 247-271

    • DOI

      10.1017/s0305004122000238

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Generalized F-signatures of Hibi rings2021

    • Author(s)
      Akihiro Higashitani, Yusuke Nakajima
    • Journal Title

      Illinois Journal of Mathematics

      Volume: 65 Issue: 1 Pages: 97-120

    • DOI

      10.1215/00192082-8827655

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Wall-and-chamber structures for algebras associated to toric cDV singularities2024

    • Author(s)
      Yusuke Nakajima
    • Organizer
      南大阪代数セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Mutations and wall-crossings for dimer models associated to toric cDV singularities2024

    • Author(s)
      Yusuke Nakajima
    • Organizer
      Advances in Cluster Algebras 2024
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] New examples of toric NCCRs arising from conic modules2024

    • Author(s)
      Yusuke Nakajima
    • Organizer
      正標数の可換環論とその周辺2024 in 淡路島
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] On crepant resolutions of toric cDV singularities2023

    • Author(s)
      Yusuke Nakajima
    • Organizer
      東京可換環論セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Wall-and-chamber structures of stability parameters for some dimer quivers2023

    • Author(s)
      Yusuke Nakajima
    • Organizer
      第55回環論および表現論シンポジウム
    • Related Report
      2023 Annual Research Report
  • [Presentation] Dimer models in representation theory (Gong show talk)2023

    • Author(s)
      Yusuke Nakajima
    • Organizer
      Silting in Representation Theory, Singularities, and Noncommutative Geometry
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Projective crepant resolutions of toric cDV singularities and dimer combinatorics2023

    • Author(s)
      Yusuke Nakajima
    • Organizer
      McKay correspondence, Tilting theory and related topics
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Dimer combinatorics for toric cDV singularities2023

    • Author(s)
      Yusuke Nakajima
    • Organizer
      特異点論月曜セミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Dimer combinatorics for 3-dimensional toric hypersurfaces2022

    • Author(s)
      Yusuke Nakajima
    • Organizer
      IPMU特異点セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] The Homological MMP for three dimensional toric hypersurfaces2022

    • Author(s)
      Yusuke Nakajima
    • Organizer
      可換環論の新しい融合セミナーII
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Combinatorial mutations and deformations of dimer models2021

    • Author(s)
      Yusuke Nakajima
    • Organizer
      Nottingham University, Online Algebraic Geometry Seminar
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Combinatorial mutations of polytopes arising from plabic graphs2021

    • Author(s)
      Yusuke Nakajima
    • Organizer
      Infinite Analysis 21 Workshop Around Cluster Algebras
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Representation theory of algebras arising from dimer models2021

    • Author(s)
      Yusuke Nakajima
    • Organizer
      OCAMI代数セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] On non-commutative crepant resolutions of toric singularities2020

    • Author(s)
      Yusuke Nakajima
    • Organizer
      阪大オンライン代数幾何学セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Combinatorial mutations of polygons via dimer models2020

    • Author(s)
      Yusuke Nakajima
    • Organizer
      McKay correspondence, mutation and related topics
    • Related Report
      2020 Research-status Report
    • Invited
  • [Remarks] Yusuke Nakajima's Web Page

    • URL

      https://sites.google.com/view/nakajima-math

    • Related Report
      2023 Annual Research Report 2022 Research-status Report 2021 Research-status Report 2020 Research-status Report

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Published: 2020-04-28   Modified: 2025-01-30  

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