Combinatorics of partially ordered sets and quantum symmetries

• Project/Area Number: 16K05083
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Meijo University
• Principal Investigator: Maeno Toshiaki 名城大学, 理工学部, 教授 (60291423)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 代数的組合せ論 / 量子代数 / 鏡映群 / Hopf代数 / 代数学 / 半順序集合

Outline of Final Research Achievements

The aim of the project is to study the structure of (finite) partially ordered sets from the viewpoints of combinatorial ring theory and of quantum deformations.
One of the main results of this project is the Lefschetz property for a certain class of Gorenstein algebras associated with matroids. This result implies the Sperner property for the geometric modular lattices. I also introduced the notion of the Solomon-Terao algebra for hyperplane arrangements with the collaborators, and have shown its basic properties.
The quantum K-theory for the flag variety was also studied in the project. The correspondence between quantum Grothendieck polynomials and K-k-Schur functions has been proved as a K-theoretic analogue of the Peterson isomorphism.

Academic Significance and Societal Importance of the Research Achievements

順序集合は理論的な考察対象としてのみならず、応用数理においても重要な役割を果たしている。本研究は、組合せ環論的アプローチや量子変形の観点から順序構造の組合せ的研究を目指したものである。
主な成果としてマトロイドから定まるGorenstein環を導入し、そのLefschetz性を研究したが、これは今後マトロイドに関わる組合せ環論の研究において基礎的な結果になるものと考えられる。
また、旗多様体の量子K環に関して量子Grothendieck多項式とK-k-Schur多項式との対応を研究したが、これはWeyl群上のBruhat順序の拡大とアフィンWeyl群上のBruhat順序の対応に相当している。

Research Products

• [Int'l Joint Research] CIRM(イタリア)
• [Int'l Joint Research] Institut Mittag-Leffler(Sweden)
• [Journal Article] Solomon-Terao algebra of hyperplane arrangements (2019)
  • Author(s): Takuro Abe, Toshiaki Maeno, Satoshi Murai and Yasuhide Numata
  • Journal Title: Journal of the Mathematical Society of Japan Volume: to appear
  • NAID: 130007733331
  • Peer Reviewed
• [Journal Article] Polynomial expressions of p-ary auction functions (2019)
  • Author(s): Shizuo Kaji, Toshiaki Maeno, Koji Nuida and Yasuhide Numata
  • Journal Title: Journal of Mathematical Cryptology Volume: to appear
  • Peer Reviewed
• [Journal Article] Peterson isomorphism in K-theory and relativistic Toda lattice (2018)
  • Author(s): Takeshi Ikeda, Shinsuke Iwao and Toshiaki Maeno
  • Journal Title: International Mathematics Research Notices Volume: - Issue: 19 Pages: 6421-6462
  • DOI: 10.1093/imrn/rny051
  • Peer Reviewed
• [Journal Article] Sperner property and finite-dimensional Gorenstein algebras associated to matroids (2016)
  • Author(s): Toshiaki Maeno and Yasuhide Numata
  • Journal Title: Journal of Commutative Algebra Volume: 8 Issue: 4 Pages: 549-570
  • DOI: 10.1216/jca-2016-8-4-549
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Presentation] Ideas and techniques in the theory of Lefschetz properties (2019)
  • Author(s): 前野 俊昭
  • Organizer: 空間の代数的・幾何的モデルとその周辺
  • Invited
• [Presentation] The Lefschetz property for Artinian Gorenstein algebras (2018)
  • Author(s): Toshiaki Maeno
  • Organizer: Algebraic Geometry in Positive Characteristic and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Polynomial expressions of auction functions (2016)
  • Author(s): Toshiaki Maeno
  • Organizer: 情報セキュリティにおける数学的方法とその実践
  • Place of Presentation: 北海道大学
  • Invited
• [Remarks] Lefschetz properties in Algebra, ...
  • URL: http://www.mittag-leffler.se/workshop/lefschetz-properties-algebra-geometry-and-combinatorics