Combinatorics of finite-dimensional algebras and quantum symmetry

• Project/Area Number: 22540015
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Meijo University (2012-2014); Kyoto University (2010-2011)
• Principal Investigator: MAENO Toshiaki 名城大学, 理工学部, 准教授 (60291423)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 代数的組合せ論 / Hopf代数 / 鏡映群 / 旗多様体 / Lefschetz性 / 量子代数

Outline of Final Research Achievements

In this project, we have obtained the results mainly on the following two themes.
(1) We have given a description of the Schubert calculus and its generalizations in terms of noncommutative differential structures on the Weyl groups. The main topics are the K-ring of the flag variety and the homology of the affine Grassmannian. We have described them as subalgebras of the Nichols-Woronowicz algebras associated with the corresponding Weyl groups.
(2) We have obtained some results on the Lefschetz property of finite-dimensional Gorenstein algebras. We have determined the set of the Lefschetz elements of the coinvariant algebra of the finite Coxeter groups (except H_4). We have also introduced a new class of Gorenstein algebras defined by matroids and proved its Lefschetz property when the matroid corresponds to a geometric modular lattice.

Research Products

• [Journal Article] A Mathematical Problem for Security Analysis of Hash Functions and Pseudorandom Generators (2015)
  • Author(s): Koji Nuida, Takuro Abe, Shizuo Kaji, Toshiaki Maeno, Yasuhide Numata
  • Journal Title: International Journal of Foundations of Computer Science Volume: 26 Issue: 02 Pages: 169-194
  • DOI: 10.1142/s0129054115500100
  • NAID: 120007100293
  • Peer Reviewed
• [Journal Article] Sperner property, matroids and finite-dimensional Gorenstein algebras. (2012)
  • Author(s): Maeno Toshiaki and Yasuhide Numata
  • Journal Title: Contemp. Math. Volume: 580 Pages: 73-84
  • DOI: 10.1090/conm/580/11496
  • Peer Reviewed
• [Journal Article] Affine nil-Hecke algebras and braided differential structure on affine Weyl groups (2012)
  • Author(s): Anatol N.Kirillov, Toshiaki Maeno
  • Journal Title: Publ.RIMS Volume: 48 Issue: 1 Pages: 215-228
  • DOI: 10.2977/prims/67
  • NAID: 110008906986
  • Peer Reviewed
• [Journal Article] A mathematical problem for security analysis of hash functions and pseudorandom generators (2011)
  • Author(s): K.Nuida, T.Abe, S.Kaji, T.Maeno, Y.Numata
  • Journal Title: Advances in Information and Computer Security, LNCS Volume: 7,038 Pages: 144-160
  • DOI: 10.1007/978-3-642-25141-2_10
  • NAID: 120007100293
  • ISBN: 9783642251405, 9783642251412
  • Peer Reviewed
• [Journal Article] Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups (2011)
  • Author(s): T.Maeno, Y.Numata, A.Wachi
  • Journal Title: Algebras and Representation Theory Volume: ((掲載確定) 電子媒体)
  • NAID: 130008004821
  • Peer Reviewed
• [Presentation] Finite geometry and the Lefschetz property (2013)
  • Author(s): Toshiaki Maeno
  • Organizer: 第35回可換環論シンポジウム
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] The Fomin-Kirillov quadratic algebra and its affinization (2013)
  • Author(s): Toshiaki Maeno
  • Organizer: Bethe Ansatz, Quantum Groups and Beyond
  • Place of Presentation: 京都大学数理解析研究所
  • Invited
• [Presentation] Alcove pathとaffine Weyl群上の非可換微分構造 (2011)
  • Author(s): 前野俊昭
  • Organizer: 日本数学会年会・無限可積分系セッション・特別講演
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-23
• [Book] The Lefschetz properties (2013)
  • Author(s): T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, J. Watanabe
  • Publisher: Springer