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Classification problem of hypergeometric differential systems

Research Project

Project/Area Number 12640149
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionHOKKAIDO UNIVERSITY

Principal Investigator

SAITO Mutsumi  Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (70215565)

Co-Investigator(Kenkyū-buntansha) SHIBUKAWA Youichi  Hokkaido Univ., Grad. School of Sci., Instr., 大学院・理学研究科, 助手 (90241299)
MATSUMOTO Keiji  Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (30229546)
YAMASHITA Hiroshi  Hokkaido Univ., Grad. School of Sci., Ass. Prof., 大学院・理学研究科, 助教授 (30192793)
WACHI Akihito  Hokkaido Inst. of Tech., Fac. of Gen. Edu., Lecturer, 総合教育研究部, 講師 (30337018)
YAMADA Hiro-fumi  Okayama Univ., Faculty of Sci., Prof., 理学部, 教授 (40192794)
三宅 敏恒  北海道大学, 大学院・理学研究科, 教授 (20025430)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordshypergeometric / toric variety / Lie algebra / classification / dimension formula / 超幾何系 / 階数公式 / 微分作用素環
Research Abstract

With support of many examples with a computer, and by communication with world-wide experts in several fields, we obtained the following results.
Mutsumi Saito generalized the classification theorem of A-hypergeometric systems to the cases when A is inhomogeneous and/or when we work in the analytic category. He also gave a dimension formula for the log-free series solutions when A is homogeneous, and a rank formula and the proof of the equivalence of Cohen-Macaulayness with the condition that the ranks are the same at all parameters, when A is homogeneous, and the convex hull of A is a simplex.
Hiroshi Yamashita obtained some results useful to know when an isotropy representation is irreducible. Furthermore he systematically constructed nonzero quotient representations of isotropy representations attached to discrete series.
Keiji Matsumoto clarified a pairing between twisted cohomology groups associated with generalized Airy functions. Writting a base of twisted cohomology groups by Young diagrams, he showed that for the base, the pairing can be explicitly written by skew-Schur polynomials.
Youichi Shibukawa solved the classification problem for R operators.
For the simplest affine Lie algebra A_1^<(1)> , using two of its realizations, Hiro-Fumi Yamada discovered the weight vectors are written by a modular version of Schur functions and Schur's Q-functions respectively.
Akihito Wachi has studied the structure of generalized Verma modules, in particular, their irreducibility, emphasizing their relations with invariant functions.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (17 results)

All Other

All Publications (17 results)

  • [Publications] Mutumi Saito: "Isomorphism classes of A-hypergeometric systems"Compositio Mathematica. 128・3. 323-338 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutumi Saito: "Differential algebras on semigroup algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules"Asterisque. 273. 81-137 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Keiji Matsumoto: "Recent progress of interesection theory for twisted (CO)-homology groups"Advanced studies in pure Math. 27. 217-237 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Youichi Shibukawa: "Classification of R-operators"J. of Math. Physics. 42. 2725-2745 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Akihito Wachi: "Capelli type identities on certain scalar generalized Verma modules"J. of math. of Kyoto Univ.. 40. 705-727 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Isomorphism classes of A-hypergeometric systems"Compositio Mathematica. 128(3). 323-338 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Differential algebras on semigroup algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalized Whittaker models for irreducible highest weight modules""Nilpotent orbits, associated cycles and Whittaker models for highest weight representations", Asterisque. 273. 81-137 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Keiji Matsumoto: "Recent of intersection theory for twisted (co) -homology group"Advanced studies in Pure Mathematics. 27. 217-237 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Youichi Shibukawa: "Classification of R-operators"Journal of Mathematical Physics. 42. 2725-2745 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Akihito Wachi: "Capelli type identities on certain scalar generalized Verma modules"Journal of Mathematics of Kyoto University. 40. 705-727 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Mutsumi Saito: "Isomorphism classes of A-hypergecmetric systems"Compositio Mathematica. 128・3. 323-338 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mutsumi Saito: "Differential algebras on semi group algebras"AMS Contemporary Mathematics. 286. 207-226 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hiroshi Yamashita: "Cayley transform and generalizecl Whittaker models for irreducible highest weight modules"Asterisgue. 273. 81-137 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Keiji Matsumoto: "Fheta constants associated with the cyclic triple coverings of the complex projective line branching at six point"Publ. RIMS. Kyoto Univ.. 37. 419-440 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Youichi Shibukawa: "Classification of R-operators"J. of Math. Physics. 42. 2725-2745 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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