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Combinatorial structures of flag varieties and quantum deformation

Research Project

Project/Area Number 19740013
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionKyoto University

Principal Investigator

MAENO Toshiaki  Kyoto University, 大学院・工学研究科, 講師 (60291423)

Project Period (FY) 2007 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥3,830,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥630,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywords旗多様体 / 鏡映群 / Hopf代数 / Lefschetz性
Research Abstract

In the present research, we revealed a relationship between a representation of the Nichols-Woronowicz algebra describing the braided differential structure on the Weyl groups and the elliptic Dunkl operators. We also constructed the Nichols-Woronowicz model of the coinvariant algebras of the complex reflection groups. Moreover, a new Pieri-type formula for the torus-equivariant (quantum) cohomology ring of the flag variety has been proved by means of an extension of the Fomin-Kirillov quadratic lgebra. In the direction of research on Lefschetz properties, we gave a characterization of he Lefschetz elements in finite-dimensional Gorenstein algebras in terms of the Hessians f the corresponding polynomials. As an application, we constructed a new series of orenstein algebras without the strong Lefschetz property.

Report

(4 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report
  • 2007 Annual Research Report
  • Research Products

    (11 results)

All 2010 2009 2008 2007

All Journal Article (11 results) (of which Peer Reviewed: 11 results)

  • [Journal Article] Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties2010

    • Author(s)
      Anatol N. Kirillov, Toshiaki Maeno
    • Journal Title

      Algebra i Analiz Vol.22

      Pages: 155-176

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Nichols-Woronowicz model of coinvariant algebra of complex reflection groups2010

    • Author(s)
      Anatol N. Kirillov, Toshiaki Maeno
    • Journal Title

      Journal of Pure and Applied Algebra Vol.214

      Pages: 402-409

    • NAID

      120002317428

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Nichols-Woronowicz model of coinvariant algebra of complex reflection groups2010

    • Author(s)
      Anatol N.Kirillov, Toshiaki Maeno
    • Journal Title

      Journal of Pure and Applied Algebra 214

      Pages: 402-409

    • NAID

      120002317428

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties2010

    • Author(s)
      Anatol N.Kirillov, Toshiaki Maeno
    • Journal Title

      Algebra i Analiz (掲載予定)

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials2009

    • Author(s)
      Toshiaki Maeno, Junzo Watanabe
    • Journal Title

      Illinois Journal of Mathematics Vol.53

      Pages: 591-603

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials2009

    • Author(s)
      Toshiaki Maeno, Junzo Watanabe
    • Journal Title

      Illinois Journal of Mathematics 53

      Pages: 591-603

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nichols-Woronowicz model of coinvariant algebra of complex reflection groups2009

    • Author(s)
      Anatol N. Kirillov and Toshiaki Maeno
    • Journal Title

      (to appear in) Journal of Pure and Applied Algebra (掲載確定)

    • NAID

      120002317428

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Braided differential structure on Weyl groups, quadratic algebras and elliptic functions2008

    • Author(s)
      Anatol N. Kirillov, Toshiaki Maeno
    • Journal Title

      International Mathematics Research Notices Vol.2008

      Pages: 23-23

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Braided allierentiat structure on Weyl groups, quadratic algebras and elliptic functions2008

    • Author(s)
      Anatol N. Kirillov and Toshiaki Maeno
    • Journal Title

      International Mathematics Research Notices 2008

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Braided differential structure on weyl groups, quadratic algebras and elliptic functions2008

    • Author(s)
      Anatol N. kirillov and Toshiaki Maeno
    • Journal Title

      International Mathematics Research Notices (掲載予定)

      Pages: 23-23

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lefsch〓tz property,Schu〓-Weyl duality and a q-deformation of Specht polynomial2007

    • Author(s)
      Toshiaki Maeno
    • Journal Title

      Communications in Algebra 35

      Pages: 1307-1321

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed

URL: 

Published: 2007-04-01   Modified: 2016-04-21  

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