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ODA Fumihito

    Department of Science　Professor
Last Updated :2024/05/14

Researcher Information

Degree

  • Doctor（Graduate School of Natural Science, Kumamoto University）
  • Master（Graduate School of Science, Hokkaido University）

URL

Research funding number

  • 00332007

J-Global ID

Research Interests

  • orthogonal unit   groupoid   locally finite category   abstract Burnside ring   Yoshida algebra   double Burnside ring   2-functor   2-category   Mackey 2-motive   Mackey 2-functor   unit group   有限群   バーンサイド環   マッキー関手   丹原関手   カテゴリー   マッキー代数   表現論   crossed Burnside ring   Green functor   表現環   ドレス構成   代数   quantum double   GBR   Drinfeld double   algebra   Mackey   centric p-radical subgroup   圏論   表現   category   フュージョンシステム   Burnside   representation   一般バーンサイド環   finite group   

Research Areas

  • Natural sciences / Algebra

Academic & Professional Experience

  • 2019/04 - Today  Kindai UniversityFaculty of Science and TechnologyProfessor
  • 2013/04 - 2019/03  Kindai UniversityFaculty of Science and TechnologyAssociate Professor
  • 2009/04 - 2013/03  Yamagata universityFaculty of SciencesAssociate Professor
  • 2009/04 - 2013/03  Yamagata UniversityFaculty of Science准教授
  • 2008/04 - 2009/03  Toyama National College of Maritime Technology一般教養学科Associate Professor
  • 2000/04 - 2008/03  Toyama National College of Technology一般科目Associate Professor
  • 1997/10 - 2000/03  日本学術振興会特別研究員(PD) （北海道大学）
  • 1997/11 - 1998/09  ミネソタ州立大学数学教室research associate
  • 1997/04 - 1997/09  日本学術振興会特別研究員(DC) （熊本大学）

Education

  • 1992/04 - 1997/09  Kumamoto University  Graduate School of Science and Technology  システム科学専攻（博士課程）
  • 1990/04 - 1992/03  Graduate school, Hokkaido University  Faculty of Science  数学専攻（修士課程）
  • 1986/04 - 1990/03  Hokkaido University of Education

Association Memberships

  • THE MATHMATICAL SOCIETY OF JAPAN   

Published Papers

  • Fumihito Oda; Yugen Takegahara; Tomoyuki Yoshida
    Algebra universalis Springer Science and Business Media LLC 81 (4) 0002-5240 2020/11 [Refereed]
  • Fumihito Oda; Masahiro Wakatake
    Hokkaido Mathematical Journal 48 (2) 345 - 356 2019/07 [Refereed]
  • FUMIHITO ODA; HIROYUKI; NAKAOKA
    Sugaku Expositions 32 (1) 57 - 86 2019/06 [Refereed][Invited]
  • F. Oda; Y. Takegahara; T. Yoshida
    Journal of Algebra 512 1 - 19 2018/10 [Refereed]
  • Tomoyuki Yoshida; Fumihito Oda; Yugen Takegahara
    Journal of Algebra Academic Press Inc. 505 339 - 382 1090-266X 2018/07 [Refereed]
     
    In this paper, we study abstract Burnside rings of essentially finite categories. Under unique epi-mono factorization property and the existence of coequalizers for some kind, we prove the existence of a fundamental exact sequence for ABR. Furthermore, an ABR can be embedded into a direct product of rational character rings.
  • Fumihito Oda; Yugen Takegahara; Tomoyuki Yoshida
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 460 370 - 379 0021-8693 2016/08 [Refereed]
     
    The unit group of a partial Burnside ring relative to the Young subgroups of the symmetric group S-n on n letters is included in the image by the tom Dieck homomorphism. As a consequence of this fact, the alternating character nu(n) of S-n is expressed explicitly as a Z-linear combinations of permutation characters associated with finite left S-n-sets S-n/Y for the Young subgroups Y. (C) 2016 Elsevier Inc. All rights reserved.
  • Hokuto Idei; Fumihito Oda
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 429 (1) 318 - 323 0021-8693 2015/05 [Refereed]
     
    In this article, we introduce a short proof of T = KC, where T is the table of marks with respect to the Young subgroups, K is the Kostka matrix, and C is the character table of the symmetric group S-n. As a corollary of the result, we could determine the explicit elements of the unit group of the generalized Burnside ring with respect to the Young subgroups of S-n. (C) 2015 Elsevier Inc. All rights reserved.
  • Oda, Fumihito; Nakaoka, Hiroyuki
    Sugaku The Mathematical Society of Japan 67 (1) 55 - 81 0039-470X 2015/01 [Refereed][Invited]
  • Fumihito Oda; Masato Sawabe
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 334 (1) 219 - 231 0021-8693 2011/05 [Refereed]
     
    In this paper, we study the generalized Burnside ring Omega(G, D) with respect to a collection D of self-normalizing subgroups. It is shown that the ordinary Burnside ring Omega (G) can be decomposed into Omega(G, D) and the kernel of a certain ring homomorphism rho(D). A basis of Ker rho(D) is also investigated. Furthermore we give a formula for the unit I(D)(G) of Omega (G, D), which is related with the Euler characteristic. As example, we take D as a collection of the normalizers of certain p-radical subgroups of G. Then the unit I(D)(G) is realized as the Lefschetz invariant of the order complex of D. (C) 2011 Elsevier Inc. All rights reserved.
  • Fumihito Oda; Tomoyuki Yoshida
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 327 (1) 31 - 49 0021-8693 2011/02 
    We discuss the Dress construction for a Tambara functor associated to a finite commutative monoid with action of a finite group. (C) 2010 Elsevier Inc. All rights reserved.
  • Fumihito Oda
    ALGEBRAS AND REPRESENTATION THEORY SPRINGER 13 (2) 231 - 242 1386-923X 2010/04 [Refereed]
     
    Let p be a prime number. This paper solves the question of the difference between the rank of the crossed Burnside ring B-c(P) of a finite p-group P and of the rational representation ring R-Q(D(P)) of the Drinfel'd double D(P) of the group algebra QP. The difference is represented by using the Dade groups of certain subgroups of P.
  • Fumihito Oda; Masato Sawabe
    ADVANCES IN MATHEMATICS ACADEMIC PRESS INC ELSEVIER SCIENCE 222 (1) 307 - 317 0001-8708 2009/09 [Refereed]
     
    In this paper, we will introduce a new collection of subgroups; which induces a generalized Burnside ring. This collection arises from the normalizers of certain p-radical subgroups (C) 2009 Elsevier Inc. All rights reserved.
  • Fumihito Oda
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 320 (10) 3726 - 3732 0021-8693 2008/11 [Refereed]
     
    Let X be the set of all p-centric subgroups of a finite group G and a prime p. This paper shows that the certain submodule Omega(G, x)((p)) of the Burnside ring Omega(G)((p)) of G over the localization Z((p)) of Z at p has a unique ring structure such that the mark homomorphism phi((p)) relative to x is an injective homomorphism. A key lemma of this paper is that x satisfies the condition (C)(p) that is discussed by [T. Yoshida, The generalized Burnside ring of a finite group, Hokkaido Math. J. 19 (1990) 509-574]. Diaz and Libman showed that certain ring A(p-cent)(G)((p)) is isomorphic to the Burnside ring of the fusion system associated to G and a Sylow p-subgroup in [A. Diaz, A. Libman, The Burnside ring of fusion systems, preprint, 2007]. This paper shows that A(p-cent)(G)((p)) is isomorphic to Omega(G, x)((p)). (C) 2008 Elsevier Inc. All rights reserved.
  • Fumihito Oda
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 315 (1) 18 - 30 0021-8693 2007/09 [Refereed]
     
    We use a formula for primitive idempotents of the crossed Burnside ring given by F. Oda and T Yoshida, and a result from the theory of Green functors obtained by S. Bouc to prove an induction theorem for the Drinfel'd double for a finite group over the complex field. We obtain Artin's induction theorem for group algebras as a corollary of the theorem. (C) 2007 Elsevier Inc. All rights reserved.
  • F Oda; T Yoshida
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 282 (1) 58 - 82 0021-8693 2004/12 [Refereed]
     
    We present some results about the Dress construction of Green functor associated to a G-monoid. In particular, we see that the crossed Burnside rings, the representation rings of quantum double and Hochschild cohomology rings are constructed by the Dress construction of some Green functors. (C) 2004 Elsevier Inc. All rights reserved.
  • F Oda; T Yoshida
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC ELSEVIER SCIENCE 236 (1) 29 - 79 0021-8693 2001/02 [Refereed]
     
    Let G be a finite group and S a finite G-monoid. A crossed G-set over S is a finite G-set equipped with a G-map into S called a weight function. A crossed Burnside ring X Ohm (G, S) is the Grothendieck ring of the category of crossed G-sets with respect to disjoint unions and tensor products. In this paper, we prove the fundamental theorem of crossed Burnside rings and an idempotent formula at characteristic 0. (C) 2001 Academic Press.
  • F Oda; T Yoshida
    JOURNAL OF ALGEBRA ACADEMIC PRESS INC 236 (1) 349 - 354 0021-8693 2001/02 [Refereed]
     
    We determine the number of blocks of the generalized Burnside ring of the symmetric group S-n with respect to the Young subgroups of S-n over a field of characteristic p. Let kS(n) be a group algebra of S-n over a field k of characteristic p > 0 and R(kS(n))((p)) the Grothendieck ring of kS(n) over p-local integers. Then, as a corollary of the theorem, we have that F x R(kS(n))((p)) is semisimple, where F is any field of characteristic p. It is well known that the result holds for an arbitrary finite group, but our approach to the result is remarkable. (C) 2001 Academic Press.
  • Primitive idempotents of the Grothendieck ring of Mackey functors
    oda fumihito
    Hokkaido Math. J. 27 (2) 383 - 392 1998 [Refereed]
  • Fumihito Oda
    Hokkaido Mathematical Journal 26 (2) 405 - 409 0385-4035 1997 [Refereed]
     
    In this paper, we introduce a new Mackey functor T and give a relation of ordinary defect group and defect group of the Mackey algebra of a finite group. © 1997 by the University of Notre Dame. All rights reserved.
  • Fumihito Oda
    Hokkaido Mathematical Journal 25 (1) 93 - 96 0385-4035 1996 [Refereed]
     
    The Burnside ring Ω(G) of a finite group G has, as an abelian group, a decomposition Ω(G)=Ω(G, χ) ⊕ K(χ) where K(χ) is an ideal and Ω(G, χ) is the generalized Burnside ring with respect to a family χ of subgroups of G. © 1996 by the University of Notre Dame. All rights reserved.

Books etc

MISC

Research Grants & Projects

  • 文部科学省：科学研究費 基盤研究（C）
    Date (from‐to) : 2019/04 -2023/03 
    Author : 小田 文仁
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2011/04 -2015/03 
    Author : FUJITA Ryousuke; SAWABE Masato; ODA Fumihito; KONO Susumu
     
    I obtained some new informations on homotopy properties of the Quillen complex of a genereal finite group, of course, including a symmetric group, from a topological point of view. As a result, it follows that the Quillen complex of a symmetric group is equivariant homotopy equivalent to the Quillen complex of its alternative group whose dimension is more than five. Besides, I contributed to create a new equivariant theory by finding the relation to the finite topological space theory.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2013 -2015 
    Author : ODA Fumihito
     
    We showed that there is a relation between the unit element of the generalized Burnside ring of a symmetric group relative to the Young subgroups, the reduced Lefschetz module and the tom Dieck homomorphism. More precisely, we characterized a non-identity unit of the generalized Burnside ring of a symmetric group relative to the Young subgroups in terms of the tom Dieck homomorphism. Consequently, we have shown that the unit group of the ring is included in the image by the tom Dieck homomorphism. We have submitted to a paper of the result to a journal of mathematics. We showed that the rank of the unit group of the generalized Burnside ring of a dihedral group relative to the parabolic subgroups is two. We have submitted to a paper of the result to a journal of mathematics.
  • Ministry of Education, Culture, Sports, Science and Technology：Grants-in-Aid for Scientific Research(基盤研究(C))
    Date (from‐to) : 2009 -2012 
    Author : Fumihito ODA
     
    (1) For a finite group and a certain family of subgroups of the group induce a categoryand a commutative ring. The ring is called generalized Burnside ring. So far only afew examples of such families of subgroups have been examined. We introduced anew family in the case when the coefficient ring is the localization of the rationalintegers at any prime. The family consists of normalizers of certain p-radical subgroups. We investigate the kernel of certain homomorphism and the units andprimitive idempotents of the ring. It is shown that the degree of the generalizedcharacter afforded by the multiplicative identity (unit) of the ring coincides withthe Euler characteristic of the order complex of the family of subgroups of the group.(2) By using a theorem of p-biset functor obtained by Bouc and Thevenaz, we have arelationship between the Dade group, crossed Burnside ring and rational representation ring of Drinfeld double for a p-group.(3) We described the construction of the crossed Burnside ring for a finite group via theDress construction applied to the Burnside Tambara functor.
  • Ministry of Education, Culture, Sports, Science and Technology：Grants-in-Aid for Scientific Research(基盤研究(C))
    Date (from‐to) : 2006 -2008 
    Author : Fumihito ODA
     
    Gを有限群とする.GをG-共役の作用でG-集合とみたものをG^cと書く.バーンサイド関手Bから表現環関手R へのマッキー関手の自然変換にG^cから得られるドレス構成法を施す.新たに得られた自然変換の1点G-集合における値が,Gの斜バーンサイド環からGのドリンフェルトダブルの表現環への自然な環準同型写像を与えているという定理を得た.さらにこの結果を応用して次の結果を得た.Gとしてp-群を考えた場合の斜バーンサイド環とドリンフェルトダブルの表現環の階数の差がデイド群の階数の和として表現できるという公式を与えた.Gのp-centric部分群の族の一般バーンサイド環は,Gのフュージョンシステムのバーンサイド環に同型であるという結果を得た.さらに,一般バーンサイド環を構成する新しい部分群の族を発見した.
  • 文部科学省：科学研究費補助金(若手研究(B))
    Date (from‐to) : 2003 -2005 
    Author : 小田 文仁
     
    今年度主に研究したことは,グリーン関手のドレス構成の応用に関することである.とくにバーンサイド環グリーン関手とグロタンディーク環(表現環)グリーン関手のドレス構成について,バーンサイド環から表現環への自然変換Fから得られる結果について研究した.有限群G自身を共役の作用でG-集合とみなしたときFのドレス構成からはクロストバーンサイド環から群環のドリンフェルトダブルの表現環への自然変換が得られる.この結果とクロストバーンサイド環のべき等元公式を用いることにより,群環のドリンフェルトダブルの表現の誘導定理を得ることができた.この定理の系として,表現環の通常の誘導定理を得ることができる.この結果の一部は,数理解析研究所講究録に掲載された.また,詳細をJournal of Algebraに投稿した.現在査読者からのレポートに従って一度目の修正作業を続行している.丹原関手は可換なグリーン関手でさらに乗法的誘導をもつ関手である.可換がG-モノイドについてグリーン関手のドレス構成で新たにグリーン関手が得られるが,さらにもとのグリーン関手が丹原関手の場合,そのドレス構成で得られたグリーン関手はまた丹原関手の構造を持つという結果が得られている.これについては,研究協力者の吉田知行(北海道大学)氏との共同研究として成果をまとめているところである.あまり,単純ではない可換図式に関する議論が必要なため,もう少し作業に時間を要する.次の研究課題として取り上げる予定である.今年度スイスのローザンヌ工科大学で開催された国際会議に参加した.情報交換に努めた結果,新たな研究の視点や課題を得ることができた.
  • 文部科学省：科学研究費補助金(奨励研究(A), 若手研究(B))
    Date (from‐to) : 2001 -2002 
    Author : 小田 文仁
     
    研究目的「(1)Crossed Burnside ringの正標数体係数のときのべき等元公式を示すこと」遂行のために,「小さな有限群に対し,標数0の場合についてそのべき等元の計算」を行った.有限群計算ソフトGAP-Groups, Algorithms, and Programming, Version 4, Aachen, St. Andrews, 1999(http://www-gap.dcs-st-and.ac.uk/gapという強力な道具を用いることにより,7次の交代群までの標数2および3の場合のべき等元の計算ができた.計算結果をそのまま論文という形で公表せずに,さらに,その結果に考察を加えた上で論文"Brauer quotients of a crossed Burnside functor"を公表することができた.この論文は2001年に出版された"Crossed Burnside rings I. Fundamental Theorems"における主定理の別証明も含む結果である.しかし,最終的な目標であるところの「正標数の体の上のべき等元公式」については,現在論文執筆中である.研究目的「(2)B型,D型のWeyl群に対してそのモジュラー指標環の係数環が正標数の体の場合に半単純となることを一般バーンサイド環を用いて示すこと」遂行のために「B型,D型の位数の小さなWeyl群について,実際にモジュラー指標環を構成」した.GAPのシェアパッケージ"CHEVIE"を利用して位数の小さい場合に構成することができた.この結果は,平成13年8月にイギリス,オックスフォード大学で行われた研究集会"Groups St. Andrews in Oxford 2001"で講演した.現在その報告集に掲載するため,査読中である.Crossed Burnside rings, generalized Burnside ringsを基礎の係数環としてとらえることができる,マッキー関手の理論に対して,一つの有限群に限定せず,有限群全体の圏からの関手とみて一般化したものをglobally defined Mackey functorという.これを道具として研究に用いるために,研究した結果を論文"Globally defined Mackey functorsについて"として公表することができた.また,平成14年6月にアメリカ,マウントホリヨーク大学でアメリカ数学会のサマーリサーチインスチチュートとして行われた研究集会"Groups, Representation and Cohomology"において, crossed Burnside ring Mackey functorsに関する研究の成果を講演した.

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