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Studies on Representations of Finite Groups and Applications

Research Project

Project/Area Number 09640063
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionSophia University

Principal Investigator

SHINODA Ken-ichi  Dept. of Mathematics, Sophia University, Professor, 理工学部, 教授 (20053712)

Co-Investigator(Kenkyū-buntansha) NAKASHIMA Toshiki  Dept. of Mathematics, Sophia University, Lecturer, 理工学部, 講師 (60243193)
WADA Hideo  Dept. of Mathematics, Sophia University, Professor, 理工学部, 教授 (10053662)
YOKONUMA Takeo  Dept. of Mathematics, Sophia University, Professor, 理工学部, 教授 (00053645)
TSUNOGAI Hiroshi  Dept. of Mathematics, Sophia University, Assistant, 理工学部, 助手 (20267412)
GOMI Yasushi  Dept. of Mathematics, Sophia University, Assistant, 理工学部, 助手 (50276515)
都築 正男  上智大学, 理工学部, 助手 (80296946)
並河 良典  上智大学, 理工学部, 助教授 (80228080)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1998: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1997: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywordscharacter sum / gaussian sum / Kloosterman sum / unitary Kloosterman sum / Gelfand-Graev representation / almost character / Hilbert scheme of G-orbit / coinvariant algebra / Character sum / unitary Kloosterman sum / Gelfand-Gracv representation / 有限簡約代教群 / 指標和 / 余不変式環 / ガウス和 / SL(3,C)の有限部分群 / 有限簡約代数群 / ヒルベルトスキーム / クルスターマン和 / 中単指標
Research Abstract

1. (A) We gave a detailed description of the fiber over the origin of Hilbert-Chow morphism from G-orbit Hubert scheme, HilbィイD1GィエD1(CィイD13ィエD1), to CィイD13ィエD1/G in the case where G is a fnite simple subgroup of SL(3, C) of order 60 or 168. This work is being clone jointly with I. Nakamura.
(B1) We defined a character sum, called a gaussian sum, over a finite reductive group and showed that if the sum is associated with the Delige-Lusztig generalized character, then it is expressed explicitly as a sum over the torus ; as an application we also showed that in the case of finite classical groups, the sum can be expressed using Kloosterman sums and unitary Kloosterman sums. N. Saito works jointly with us on this problem.
(B2) We investigated Hasse-Davenport type relation for Kloosterman sums and unitary Kloosterman sums ; we also applied it to the norm map of Gelfand-Graev representation of GL (2, q) ; jointly studied with C. W. Curtis (Oregon U.)
c We determined the values of 7 unipotent almost characters of finite Chevalley groups of type FィイD24ィエD2 without assumuptions on characteristic of the ground field ; some of them were known already.
2. We showed that the Hasse princile holds for symmetric groups and alternating groups, etc. (Wada ; joint with T. One)
3. We gave a method, called the polyhedral realization, to describe the crystal base associated with aim irrreducible highest weight module of a quantum group. (Nakashimna)
4. We classified the case when the parabolic Heck algebras remain semisimple after specializing over arbitrary field. (Gomi)
5. From the standpoint of Number Theory, we investigated the ranks on the stable derivation algebra related with Dehigne's problem (Tsunogai) and also zeta functions associated with the Riemmiannian symmetric space of rank 1 and the discrete coconipact automorphism subgroup. (Tsuzuki)

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report

Research Products

(54 results)

All Other

All Publications (54 results)

  • [Publications] K. Shinoda: "Values of the unipotent characters of the Chevalley group of type F_4"Proc. of Alg. Comb. and related topics, Manila 1994. 113-119 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] C. W. Curtis: "Unitary Kloosterman sums and the Gelfand-Graev representation of GL_2"J. of Algebra. 216. 431-447 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] N.Saito: "Character sums attached to finite reductive groups"Tokyo J. of Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Gomi: "Hilbert schemes of G-orbits in dimension three"Asian J. of Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ono: ""Hasse principle" for free groups"Proc. Japan Academy. 75. 1-2 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ono: ""Hasse principle" for symmetric and alternating groups"Proc. Japan Academy. 75. 61-62 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Wada: ""Hasse principle" for SL_n(D)"Proc. Japan Academy. 75. 67-69 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Nakashima: "Polyhedaral realizations of crystal bases for integrable heighest weight modules"Journal of Algebra. 219. 571-597 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Nakashima: "Polyhedaral realizations of crystal bases and braid-type isomorphisms"Contemporary Math.. 248. 419-435 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Gomi: "Semisimplicity of Parabolic Hecke Algebras"J. of Algebra. 203. 270-284 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Tsunogai: "On ranks of the stable derivation algebra and Deligne's problem"Proc. Japan Academy. 73. 29-31 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Hashimoto: "On the Sato-Tate conjecture for QM-curves of genus two"Math. of Computation. 68. 1649-1662 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Tsuzuki: "Elliptic factors of Selberg zeta functions"Duke Math. J.. 88. 29-75 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Tsuzuki: "Real Shintani functions and multiplicity free property for the symmetric pair (SU(2,1),S(U(1,1)×U(1)))"J. Math. Sci. Univ. Tokyo. 4. 663-727 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shioda, Ken-ichi: "Value of the unipotent characters of the Chevalley group of type FィイD24ィエD2"Proceedings of Algebraic Combinatorics and related topics, Manila 1994. 113-119 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shinoda, Ken-ichi, Charles W. Curtis: "Unitary Kloosterman sums and Gelfand-Graev representation of GLィイD22ィエD2"J.Algebra. 216. 431-447 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shinoda, Ken-ichi, Naomichi Saito: "Character sums attached to finite reductive groups"Tokyo J.Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Shinoda, Kenichi, Yasushi Gomi and Iku Nakamura: "Hilbert Schemes of G-Orbits in Dimention Three"Asian J.Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Wada, Hideo, Takashi Ono: ""Hasse principle" for free groups"Proc.Japan Acad.. 75A. 1-2 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Wada, Hideo, Takashi Ono: ""Hasse principle" for symmetric and alternating groups"Proc.Japan Acad.. 75A. 61-62 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Wada, Hideo: ""Hasse principle" for SLィイD2nィエD2 (D)"Proc.Japan Acad.. 75A. 67-69 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakashima, Toshiki: "Polyhedral Realizations of Crystal Bases for Integrable Highest Weight Modules"Journal of Algebra. 219. 571-507 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Nakashima, Toshiki: "Polyhedral Realizations of Crystal Bases and Braid-type Isomorphisms"Contemporary Mathematics. 248. 419-435 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Gomi, Yasushi: "Semisimplicity of Parabolic Hecke Algebras"J.Alg.. 203. 270-284 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tsunogai, Hiroshi: "On ranks of the stable derivation algebra and Deligne's problem"Proc.Japan Acad.. 73A. 29-31 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tsunogai, Hiroshi, K. Hashimoto: "On the Sato-Tate Conjecture for QM-curves of genus two"Math.Comp.. 68. 1649-1662 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tsuzuki, Masao: "Elliptic factors of Selberg zeta functions"Duke Mathematical Journal. 88 No.1. 29-75 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tsuzuki, Masao: "Real Shintani functions and multiplicity free property for the symmetric pair"J.Math.Sci.Univ.Tokyo. 4. 663-727 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] C.W.Curtis: "Unitary Kloosterman sums and the Gelfand-Graev representation of GL_2"Journal of Algebra. 216. 431-447 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] N.Saito: "Character sums attached to finite reductive groups"Tokyo J.of Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Gomi: "Hilbert schems of G-orbits in dimension three"Asian J.of Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Ono: ""Hasse Principle"for symmetric and alternating groupa"Proc.Japan Academy. 75. 61-62 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Ono: ""Hasse Principle"for free groups"Proc.Japan Academy. 75. 1-2 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Wada: ""Hasse Principle"for SL_n(D)"Proc.Japan Academy. 75. 67-69 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Nakashima: "Polyhedaral realizations of crystal bases for integrable heighest weight modules"Journal of Algebra. 219. 571-597 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Nakashima: "Polyhedaral realizations of crystal bases and braid-type isomorphisms"Contemporary Math.. 248. 419-435 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Gomi: "Semisimplicity of parabolic Hecke algebras"Journal of Algebra. 203. 270-284 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Hashimoto: "On the Sato-Tate conjecture for QM-curves of genus two"Math.of Computation. 68. 1649-1662 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Tsunogai: "On ranks of the stable derivation algebra and Deligne's problem"Proc.Japan Academy. 73. 29-31 (1997)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Tsuzuki: "Elliptic factors of Selberg zeta functions"Duke Math J.. 88. 29-75 (1997)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Tsuzuki: "Real Shintani functions and multiplicity free property for the symmetric pair (SU(2,1),S(U(1,1)xU(1)))"J.Math.Sci.Univ.Tokyo. 4. 663-727 (1997)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Yokoyama: "Introduction to Lie Algebras,JSPS-DOST,Lecture note in mathematics vol.6"Sophia Univ.. 84 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] C. W. Curtis: "Unitarg Kloosterman Sums and Gelfand-Graev Representation of GL_2" J. Algebra. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] Y. Gomi: "Semisimplicity of Parabolic Hecke Algebras" J. Algebra. 203. 270-284 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T. Nakashima: "Polyhedral realigations of crystal bases for integrable highest weight modules" J. Algebra. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] M. Tsuzuki: "Elliptic factors of selberg zeta functions" Duke Math. J.88. 29-75 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] K. Hashimoto: "On the Sato-Tate conjecture for QM-curves of genus two" Math. in Computation. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] T. Ono: ""Hasse principle" for free groups" Proc. Japan Acad.75. 1-2 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.SHINODA: "Values of the unipotent cheracters of the Chevalley gwoup of type F_4" Algebreoic Coubinaforics and Relateal Topics,Manila 1994. 113-119 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.GOMI: "Semisimplicity of parabolic Hecke algcbros" J.Alg.(to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.NAMIKAWA: "Smoothing Faro 3-folds" J.Alg.geom.(to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] H.TSUNOGAI: "On ranks of the stable derivation algebra and Deligne's problem" Proc.Japan Acad.73A. 29-31 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] M.TSUZUKI: "Elliptic factors of Selberg zeta functions" Duke Math.J.(to appear).

    • Related Report
      1997 Annual Research Report
  • [Publications] K.SHINODA: "Coinvariant algebras of some finite groups" 数理解析研究所講究録. 991. 137-140 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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