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KAWATA Shigeto

    Graduate School of Science Division of Mathematical and Material Science　Professor
Contact： kawatansc.nagoya-cu.ac.jp
Last Updated :2024/09/26

Researcher Information

URL

Research funding number

  • 50195103

J-Global ID

Research Interests

  • 表現論   有限群   

Research Areas

  • Natural sciences / Algebra

Association Memberships

  • 日本数学会   

Published Papers

Conference Activities & Talks

  • Tensor products and almost split sequences
    Shigeto Kawata
    The Ninth China-Japan-Korea International Conference on Ring and Module Theory (Incheon National University, 韓国)  2023/08
  • 群環上のScott加群とテンサー積について
    河田成人
    2021年度日本数学会秋季総合分科会  2021/09
  • 群環の概分裂完全列とテンサー積について
    河田成人
    2019年度日本数学会秋季総合分科会  2019/09
  • 群環上の直既約加群のヴァーテックスについて
    河田成人
    2017年度日本数学会秋季総合分科会  2017/09
  • 群環の Auslander-Reiten連結成分とヴァーテックス
    河田成人
    2016年度日本数学会秋季総合分科会  2016/09
  • 有限群のブロックにおける高さ0の表現加群とAuslander-Reiten連結成分について
    河田成人
    2015年度日本数学会秋季総合分科会  2015/09
  • 群環のAuslander-Reiten成分とテンサー積について
    河田成人
    2011年度日本数学会秋季総合分科会  2011/09
  • 群環におけるAuslander-Reiten成分とトレース写像が分裂する加群について
    河田成人
    2010年度日本数学会秋季総合分科会  2010/09
  • 群環の自明なソースを持つ加群とAuslander- Reiten quivers
    河田成人
    2009年度日本数学会秋期総合分科会  2009/09
  • Heller lattices and Auslander-Reiten quivers for integral group rings
    Shigeto Kawata
    XII International Conference on Representations of Algebras and Workshop (Nicolaus Copernicus University, Torun, Poland)  2007/08
  • 群環のHeller格子の直既約性について
    河田成人
    2005年度日本数学会秋季総合分科会  2005/09
  • On Auslander-Reiten components for integral group rings of p-groups
    Shigeto Kawata
    The 9th International Conference on Representations of Algebras (Beijing Normal University)  2000/08
  • On Auslander-Reiten components and simple modules for group algebras
    Shigeto Kawata
    “Representation theory of finite groups” Conference in Oberwolfach  1996/04
  • Module correspondence in Auslander-Reiten quivers for finite groups
    Shigeto Kawata
    International Conference on Representation Theory of Groups and Related Topics (The University of Manchester Institute of Science and Technology，England)  1988/07

MISC

  • Knorr格子とテンサー積について
    河田 成人  数理解析研究所講究録2252 「有限群のコホモロジー論とその周辺」  15  -26  2023/05
  • Scott加群の概分裂完全列とテンサー積について
    河田 成人  数理解析研究所講究録2134 「有限群のコホモロジー論とその周辺」  45  -51  2019/11
  • 群整環上の表現加群のヴァーテックスについて
    河田 成人  数理解析研究所講究録2061 「有限群のコホモロジー論とその周辺」  48  -55  2018/04
  • 群環の表現加群のヴァーテックスとAuslander-Reiten連結成分について
    河田 成人  数理解析研究所講究録2053 「有限群・代数的組合せ論・頂点作用素代数の研究」  111  -118  2017/10
  • 河田 成人  数理解析研究所講究録1872「 有限群とその表現,頂点作用素代数,代数的組合せ論の研究」  140  -150  2014/01
  • 群環のHeller格子について
    河田 成人  数理解析研究所講究録1564 「群論とその周辺」  70  -75  2007/07
  • 群環の自明なソースをもつ加群とAuslander-Reiten列について
    河田 成人  数理解析研究所講究録1251「有限群のコホモロジー論の研究」  130  -138  2002/02
  • 群環の射影加群とAuslander-Reiten列について
    河田 成人  数理解析研究所講究録1140「有限群のコホモロジー論の研究」  80  -85  2000/04
  • KAWATA Shigeto  RIMS Kokyuroku  34  -40  1994/06
  • KAWATA Shigeto  数理解析研究所講究録799「Representation Theory of Finite Groups and Finite Dimensional Algebras」  32  -45  1992/08

Research Grants & Projects

  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    Date (from‐to) : 2019/04 -2023/03 
    Author : 河田 成人
     
    Gで有限群を表し、pは有限群Gの位数を割り切る素数とする。完備離散付値環Rの極大イデアルはπで生成され、その剰余体k = R/πRは標数がpであるとする。有限群Gの完備離散付値環R上の整数表現において、Scott 加群と呼ばれるRG-加群は重要な役割を果たす。Gのp-部分群Qに対し、Qをヴァーテックスとして持つScott RG-加群Sとは、自明なRQ-加群を誘導して得られる置換RG-加群の直既約因子であって、そのheadとsocleが自明な加群となっているものである。なお、直既約RG-表現加群Vに対し、Gの部分群の集合 { H≦G : VはH-射影的 } には極小なものが一意的に存在するが、この極小部分群をVのヴァーテックスと呼ぶ。直既約加群を点とし既約写像を矢と見なして構成されるAuslander-Reiten有向グラフにおいて、Sを含む安定連結成分Θは半平面的に無限に広がる格子型の形状をしており、SはΘの端点に位置していることが知られていた。このΘについて考察を進め、特に、Θを構成する直既約RG-表現加群LがΘの端点に位置しなければ、LのヴァーテックスはQのGにおける正規化群のSylow p-部分群であることを示した。さらに、kG-加群L/πLの直既約分解において、全ての直既約因子はQ-射影的であり、その中の少なくとも一つの直既約因子のヴァーテックスはQであることも確かめた。また、Scott RG-加群Sで終わるRG-表現加群の概分裂短完全列とScott kG-加群S/πSで終わるkG-表現加群の概分裂短完全列について並行して調べ、それらの構造や性質および類似性などについて研究した。
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    Date (from‐to) : 2014/04 -2018/03 
    Author : KAWATA Shigeto; KANEDA Masaharu; FURUSAWA Masaaki; BABA Yoshitomo
     
    Let G be a finite group and R a complete discrete valuation ring with residue class field k of positive characteristic. Suppose that a block B of the group ring RG is of infinite representation type. Let L be an indecomposable B-lattice, and let C be the stable component of the Auslander-Reiten quiver of B containing L. Assume that the reduced kG-module M of L is indecomposable. Then, we have proved that the tree class of C is A-infinity if L is of height 0. Also, we have shown that if L and M have the same vertex Q, then the vertex of C is Q.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2011/04 -2015/03 
    Author : KANEDA masaharu; TANISAKI Toshiyuki; YAGITA Nobuaki; TEZUKA Michishige; FURUSAWA Masaaki; HASHIMOTO Yoshitake; KAWATA Shigeto
     
    Let G be a reductive algebraic group over an algebraically closed field of positive characteristic p, P a parabolic subgroup of G, and T a maximal torus of P, G_1 the Frobenius kernel of G. In joint work with Abe Noriyuki we determined the G_1T-structure of G_1P-Verma modules of p-regular highest weights for large p. In joint work with H.H. Andersen we determined the cohomology vanishing behavior of line bundles on G/B in type G_2 when the corresponding B-modules lie in the lowest p^2-alcoves. In join work with M. Gros we constructed a Frobenius splitting on a quantum group at a root of unity.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    Date (from‐to) : 2011 -2013 
    Author : KAWATA Shigeto; KANEDA Masaharu; BABA Yoshitomo
     
    Let RG be an integral group ring of a finite group G over a complete discrete valuation ring R with residue class field k of positive characteristic. Suppose that L is an indecomposable RG-lattice of height zero, and let S be a source of L. We have shown that the tree class of the Auslander-Reiten component containing L is A-infinity if and only if the tree class of the Auslander-Reiten component containing S is A-infinity. Also, we have proved that the middle term of the almost split sequence terminating in L is indecomposable if the reduced kG-module of L is indecomposable. In the case of 2-modular system, we have shown that the tree classes of the Auslander-Reiten components containing odd rank RG-lattices are A-infinity.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2007 -2010 
    Author : KANEDA MASAHARU; TANISAKI Toshiyuki; YAGITA Nobuaki; TEZUKA Michishige; FURUSAWA Masaaki; HASHIMOTO Yoshitake; KAWATA Shigeto; ASASHIBA Hideto
     
    Let G be a reductive algebraic group over a field of positive characteristic, and P a parabolic subgroup of G with the respective Weyl groups W and W_P. In joint work with Ye Jiachen we have constructed a Karoubian completeexceptional sequence of coherent sheaves E_w, w/in W/W_P, on G/P in case G is of rank atmost 2,using the representation theory of G_1P, G_1 the frobenius kernel of G.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    Date (from‐to) : 2007 -2010 
    Author : KAWATA Shigeto; KANEDA Masaharu; ASASHIBA Hideto; KADO Jiro
     
    We gave a proof of the indecomposability for certain Heller lattices over integral group rings. Also, we investigated the Auslander-Reiten quiver of an integral group ring, and we showed that Auslander-Reiten components containing trivial source lattices belonging to a block of infinite representation type are of type A-infinity and trivial source lattices lie at the end of their components.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2006 -2008 
    Author : OSHIRO Kiyoichi; YOSHIMURA Hiroshi; IIYORI Nobuo; KOSHITANI Shigeo; NISHIDA Kenji; HANAKI Akihide; KOIKE Kazutoshi; KUTAMI Mamoru; KIKUMASA Isao; YAMAGTA Kunio; SATOU Masahisa; ASASHIBA Hideto; KAWATA Shigeto; BABA Yositomo; MARUBAYASI Hidetoshi
     
    研究代表者のHarada 環の構造論、Nakayama環のclassification, skew matrix ringの基礎理論を広範な応用を視野に入れて研究し、より進化した形の理論に仕上げることができた。これらの理論を基盤にして分担者の馬場との共同で"Classical Artinian Rings and RelatedTopics"なる専門書を作成した。この本は2009 年にシンガポールのWSPC社から出版されることになった。この本の完成により本研究も研究代表者の長年の研究も完結をみた
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2005 -2008 
    Author : ASASHIBA Hideto; KANEDA Masaharu; SUMIOKA Takeshi; KAWATA Shigeto; KADO Jiro
     
    可換環k上の線型圏に関する, 群の自由作用を仮定しない, 一般的な被覆理論を作り, 導来同値のための被覆理論を精密化した.その応用として, 2つの圏R, S に群G が作用するとき, Rに対するG 安定な傾部分圏E が存在して, E とS の間に弱G 同変な圏同値があれば, R とSは導来同値になることを証明した.さらにこの被覆理論を深化させCohen-Montgomery 双対性の圏論的一般化を行った.また, domestic 標準多元環による単純リー代数の実現を精密化し, 論文の初版の誤りを根本的に正して実現を完成させた.交付
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2004 -2006 
    Author : FURUSAWA Masaaki; KANEDA Masaharu; KAWATA Shigeto; KADO Jiro; ICHINO Atushi; TANISAKI Toshiyuki
     
    We have continued the projects concerning the automorphic L-functions of gereral symplectic and unitary groups of rank two. More specifically one of the main projects is to prove the generalization of Siegfried Boecherer's conjecture concerning the central critical values of the degree four L-functions for the Siegel eigen cusp forms of degree two. Our method is to establish certain relative trace formulas, which may be regarded as natural generalizations of Jacquet's relative trace formulas which have given another proof of celebrated Waldspurger's theorem on the relation between the torus period for GL(2) and the central critical values of automorphic L-functions for GL(2). In order to establish a relative trace formula, proving the fundamental lemma is the first and crucial step. We have proved the fundamental for the unit element of the Hecke algebra already and published the result as No. 782 of the Memoirs of the AMS. During the period supported by this grant, we worked on extending the fundamental lemma from the unit element to the entire Hecke algebra. We have discovered that, by applying the theory of Macdonald polynomials to the explicit formulas for the Bessel model, the evaluation of the Kloosterman orbital integral for the general element in the Hecke algebra is reduced to the computation of general Kostka numbers and that of degenerate Kloosterman orbital integrals for the unit element of the Hecke algebra. We have evaluated all of them. Now our remaining task is to compare the linear combinations of these corresponding to the both sides of the trace formula and to make sure they match.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2004 -2006 
    Author : KANEDA Masaharu; TANISAKI Toshiyuki; YAGITA Nobuakl; HASHIMOTO Yoshitake
     
    After the spectacular success of Bezrukavnikov, Mirkovic and Rumynin, BMR for short in what follows, in extending the localization theorem of D-modules on the flag variety to positive characteristic, we started to investigate the localization of ＼bar D-modules. On a smooth variety X in positive characteristic BMR's ＼mathcal{D}, which they call the sheaf of crystalline differential operators, is Berthelot's sheaf of arithmetic differential operators of level 0. In "Localization of ＼bar D-modules" written with Hashimoto and Rumynin we first gave a simple presentation of ＼mathcal{D}-{(m)} the sheaf of arithmetic differential operators of level m. After BMR we proved that each＼mathcal{D}"{(m)} is Azumaya. ＼bar ＼mathcal{D"{(m) } } is the endomorphism ring of the structure sheaf of X over its (m+1)-st Probenius twist, and is a central reduction of ＼mathcal{D"{(m) } }. We observed that the triangulated localization theorem for ＼bar＼mathcal{D}-{(m)} holds almost iff the direct image F"{m+1}_* ＼mathcal{O}_X of the structure sheaf under the (m+1)-st Frobenius morphism is tilting, and verified that on the projective space F"{m+1}_*＼mathcal{O}_X is tilting if the characteristic is large enough. On the flag variety, as the direct image of the ＼bar＼mathcal{D*{(m) } } is the whole of the sheaf of classical differential operators ＼mathcal {Diff} and as the cohomology vanishing of ＼mathcal {Diff} fails in the case of SL_5 by Kashiwara-Lauritzen, one cannot expect the triangulated localization theorem holds for all ＼bar＼mathcal{D-{(m)}1. Nevertheless, in view of BMR there may something special happening when m-0, and indeed, we found that F_*＼mathcal{O}_{G/B} is tilting in the case of SL 3 in sufficiently large characteristic. In a joint paper with Ye "Equivariant localization of ＼bar D modules on the flag variety of the symplectic group of degree 4" we also verified that F_*＼mathcal{O}_{G/B} is tilting insufficiently large characteristic in case Sp_4 also.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2002 -2004 
    Author : SUMIOKA Takeshi; TSUSHIMA Yukio; ASASHIBA Hideto; KAWATA Shigeto; KADO Jiro
     
    1. We obtained the following result on derived equivalences between self-injective algebras of the form Λ = A/ for some finite-dimensional algebras A over an algebraically closed field and some non-negative automorphism g of the repetition A of A with g^2 the Nakayama automorphism of A : A is expressed as a triangular matrix algebra (A_g__ 0__) ; and for another algebra Π = B/ of the same type, under a suitable codition on A and B, the algebras Λ and Π are derived equivalent if there is a tilting triple (Ag,T_0,B_h) such that (A,T,B) is also a tilting triple, where we put T = (T_0 【cross product】_ ε_1A) 【symmetry】 (T_0 【cross product】_ ε_2A), ε_1 := (1__0 0__0) ; and ε_2 := (0__0 0__1) ∈ A. 2. Using the Hall algebra defined by the nilpotent modules over the path-algebra of a cyclic quiver, we realized special and general linear Lie algebras. 3. We realized all types of simple complex Lie algebras as some factor Lie algebras of degenerate composition Lie algebras constructed from the Hall algebras of tame hereditary algebras. 4. We realized simple complex Lie algebras with simply-laced Dynkin diagrams Δ as some factor Lie algebras L(A) of degenerate composition Lie algebras constructed from the Hall algebras of canonical algebras A of type Δ. In addition, we constructed a Lie algebra analogous to L(A) from the isoclasses of indecomposable objects of the quotient category D^b(mod A)/ of the bounded derived category by the shift T using triangles instead of exact sequences. 5. We generalized the construction method of Lie algebras using canonical algebras in the above to construct a Lie algebra L(B) from an algebra B derived equivalent to a hereditary algebra. It is still under investigation whether this Lie algebra is invariant under derived equivalences.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
    Date (from‐to) : 2001 -2003 
    Author : KAWATA Shigeto; ASASHIBA Hideto; TAKESHI Sumioka; TSUSHIMA Yukio; KADO Jiro
     
    The purpose of this research project is to apply the Auslander-Reiten theory for Artin rings and orders to the representation theory of finite groups. First, we considered the shapes of the Auslander-Reiten components of group rings over complete discrete valuation rings and showed that those components containing the projective lattices are of the form ZA_<∞> if the groups are of prime power order. Moreover, we showed that trivial source lattices lie at the ends of their Auslander-Reiten components. Consequently, we obtained another proof of a theorem of Heller-Reiner, which asserts that the group rings of finite groups of prime cubed order over complete discrete valuation rings are of infinite representation type. On the other hand, Tsushima got some results concerning on the Hecke algebras of symmetric groups by applying the deep results in the representation theory of the symmetric groups. Also, Asashiba obtained some equivalent conditions which implies that twisted multifold extensions of piecewise hereditary algebras of tree type are derived equivalent. Furthermore, he realized general and special linear algebras via Hall algebras of cyclic quiver algebras.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 2000 -2002 
    Author : YAMAGATA Kunio; YOSHINO Yuji; MAEDA Hironobu; WADA Tomoyuki; KAWATA Shigeto; TSUSHIMA Yukio
     
    We studied representations of finite dimensional algebars over a field. In particular, we concentrated on the research of selfinjective algebas (ie Frobenius algebras) based on the theory of socle deformation studied by the joint work with A. Skowronski and the head investigator. We got the following three main results. 1. We studied about the open problem called 'ZA_∽ problem' which states that an algebra with AR-components of ZA_∽ type is wild. We proved a therem which implies, as corollaries, many known sufficient conditions for algebras to have AR-components of ZA_∽ type. In particular, in the case when an algebra has no identity, we found a counterexample of the problem. 2. We determined a structure of the rigid automorphism group of the repetitive algebras by finite dimensional algebras. 3. We studied the module categories of selfinjecitve algebras, and we proved that a selfinjective algebra A stably equivalent to an algebra B with Galois covering by a repetitive algbra has also a Galois covering by a repetitive algebra. Moreover, we determined algebras with at least three generalized standared components, and algebras of Euclidean type.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1999 -2000 
    Author : TSUSHIMA Yukio; KAWATA Shigeto; ASASHIBA Hideto; KANEDA Masaharu; WATANABE Atumi; KOSHITANI Shigeo
     
    (1) Study of endomorphism rings of permutatuion modules over finite groups In connection with the Iwahori-Hecke algebras, Tsushima has established some results on the modular representations of symmetric groups. In particular he constructs some simple constituents of the Specht modules using the operation on the Young diagram called branch. To be precise, Carter-Payne's theorem which is known to be true only for bar branch type is extended to pillar branch type. Also it is shown that each Specht module has simple constituents whose corresponding Young diagrams are branches of the original Young diagram. Moreover a complete proof has been given to the Nakayama conjecture for the q-Schur algebras, which is done because the original proof to the conjecture given by James and Mathas contains a gap. (2) Study of indecomposable modules of finite groups Watanabe has shown that Alperin conjecture is true for the principal p-block if the group under consideration has an abelian Sylow p-subgroup with automizer of prime order, which induces the validity of Broue's conjecture on perfect isometry. (3) Representation theory of groups of Lie type Kaneda has established the quantum analogue of Andersen-Haboush's theorem on the cohomology group of the simply connected simple algebraic groups, which yields at the same time the quantum analogue of Kempfs vanishing theorem.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1998 -1999 
    Author : SUMIOKA Takeshi; KAWATA Shigeto; ASASHIBA Hideto; TSUSHIMA Yukio; OKUYAMA Tetsuro; KADO Jiro
     
    In ring and representation theory, Morita duality is applied in various field and is a very important research task. In 1969, as a detail version of Morita duality, Fuller gave characterizations of indecomposable indicative ideals over right artinian rings with a relation of two projective ideals, and in1992, Baba and Oshiro extended these results to semiprimary rings. In our researches, we extended some results by Fuller and Baba-Oshiro related to projective ideals to a theory for modules by using a notion "pairs of modules" which was introduced by Morita and Tachikawa. Applying these results, we gave a condition for modules in pairs with annihilator condition to have finite Goldie dimension and gave a characterization for finitely cogenerated injective modules. These results not only extend projective ideals to modules but also clarify essence of properties, and more developments are expected. On the other hand, the Auslander-Reiten theory is one of important tools in studying the representation theory of Artin algebras. In order to apply this Auslander-Reiten theory for the representation theory of finite groups, we have considered Auslander-Reiten quivers of finite groups. In 1995, Erdmann proved that if the block of a finite group over a field is of wild representation type, then any connected component of the stable Auslander-Reiten quiver of this block has tree class AィイD2∝ィエD2. In this project we have showed that if the group ring of a finite p-group over a complete discrete valuation ring is of wild representation type, then the tree class of the connected component of the stable Auslander-Reiten quiver of this group ring containing the trivial lattice is AィイD2∝ィエD2. Also we obtained some relation between almost split sequences in the case of modular representation and those in the case of integral representation.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1998 -1999 
    Author : TAMAGATA Kunio; YOKOTE Ichiro; TASHIRO Yoshiaki; WADA Tomoyuki; KAWATA Shigeto; MAEDA Hironobu
     
    We studied representations of finite dimensional algebars over a field. In particular, we concentrated on the research of selfinjective algebas (ie Frobenius algebras). We got two main results. One is about a construction of symmetric algebras, and the other is a characterization of symmetric algebras induced from repetitive algebras by positive automorphism : (1) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we constructed a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles, and we showed a criterion condition on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles. (2) Let BィイD4^ィエD4 be the repetitive algebra of a finite dimensional algebra B over a field K by the standard duality module over B, and let ν be the Nakayama automorphism of BィイD4^ィエD4. We determined the positive automorphisms ψ of BィイD4^ィエD4 such that the orbit algebra BィイD4^ィエD4/(ψν) is isomorphic to a splittable extension algebra of B by a minimal injective cogenerator, and we characterized weakly symmetric algebras and symmetric algebras, of the form BィイD4^ィエD4 /(ψν) with a positive automorphism ψ of BィイD4^ィエD4. As an application, we characterized some class of weakly symmetric algebras with non-periodic generalized standard Auslander-Reiten components.
  • Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research
    Date (from‐to) : 1997 -1998 
    Author : TSUSHIMA Yukio; OKUYAMA Tetsurou; YAMAGATA Kunio; KAWATA Shigeto; ASASHIBA Hideto; KANEDA Masaharu
     
    (1) Indecomposable modules over finite groups Some contributions to the solution of the Broue conjecture are made especially in linear groups and the Glauberman correspondence cases. The tree classes of some Auslander-Reiten components for integral group rings of non cyclic p-groups are determined. (2) Endomorphism rings of permutation modules The orthgonality relation of characters of blocks of finite groups is generalized to that of blocks of Hecke algebras in the case where the base subgroup under consideration has order prime to the characteristic of the base field. Some new results are obtained about the number of irreducible modular constituents of Specht modules of the symmetric group. (3) Representation theory of groups of Lie type Some foundations on the theory of D-modules in positive characteristic are established. A new approach is made using Kashiwara's crystal basis to Mathieu's theorem concerning the tensor products of modules with good filtrations.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1995 -1995 
    Author : 金信 泰造; 西尾 昌治; 河田 成人; 大嶋 秀明; 枡田 幹也; 河内 明夫
     
    1980年代の中頃から終わりにかけて,JonesやWittenらによって,コンパクトLie群に付随した3次元多様体の量子不変量が定義された.これはAtiyahらによって,(2^+1)次元位相的場の理論の枠組みで捉えられようとしており,理論物理との関係が注目されている.しかしながら,その定義があまりに抽象的すぎて実際の計算に適さない.本研究の目的は具体的にLiE群や3次元多様体が与えられたときの量子不変量を具体的に計算する方法をみつけだして実際に行なうことにより,量子不変量の位相的な性質を解明することにあった.特に,SU(2)に付随した量子不変量は,3次元多様体の枠付き絡み目表示を使って計算できる.これに関して具体的な計算機実験をパソコンの数式処理ソフトにより行ない,現在も進行中である. これと並行してVassilievによって始められた結び目の位相不変量であるVassiliev不変量の研究を行った.実際,量子不変量はVassiliev不変量として捉えることもできる.最近では3次元多様体の不変量として捉えることもできる.最近では3次元多様体の不変量としても拡張されている.HOMFLY多項式がVassiliev不変量においてどれくらい寄与するか,すなわち,いわゆるHOMFLY次元に関する研究を行った.さらに,位数5以下の結び目のVassiliev不変量の基底をConway多項式,HOMFLY多項式,Kauffman多項式を使って具体的に求めた.さらに,空間グラフであるテ-タ(θ)曲線の位数3のVassiliev不変量の基底を与えた.これについては,4項関係式の他に,グラフであることから,3項関係式がでてくるが,結局は,θ曲線に一意的に対応する3成分からなる絡み目の多項式不変量を用いてVassiliev不変量の基底が得られた.この際にも,パソコンを大いに活用した.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1994 -1994 
    Author : 河田 成人
     
    有限群の表現論の研究において,いわゆるAuslander-Reitenの理論が応用されだした.この理論は多元環の表現論を研究するときに大きな威力を発揮してくる.多元環は表現論的には有限表現型,tame表現型,wild表現型とに大別されるが,Auslander-Reitenの理論を駆使することにより,有限表現型の多元環についてはほぼ満足のいく結果が得られており,tame表現型の多元環についてもかなりの進展がみられている.そして現在は,wild表現型の多元環についての研究の端緒が見い出されようとしている段階である.ところで,多元環の中でも重要な位置を占める群多元環は,多くの場合,wild表現型であり,その表現の研究は注目されている. Auslander-Reitenの理論において,まず着目すべきことは,Auslander-Reiten quiverと呼ばれる有向グラフである.このAuslander-Reiten quiverは,直既約加群を点とし,既約写像を矢とするものであるが,このグラフを考察することにより,多元環の研究が進められる.ところで最近,K.Erdmannによって,群多元環のAuslander-Reiten quiverについて重要な結果が得られた.それは,群多元環がwildであればAuslander-Reiten quiverの連結成分(AR-component)の形は‘tube'か‘ZA_∞'に限るという事実である.本研究においては,このErdmannの重要な結果を踏まえて,有限群の表現の研究を押し進めた.特に既約加群のAR-componentの中における位置に注目して,既約加群と主直既約加群との関係を考察した.まだ満足のいく結果には遠いが,しかしある条件の下では既約加群はAR-componentの中で‘端'に位置することが分かった.今後は,もっと一般的に既約加群のAR-componentの中における位置について考察を進めることが課題とされる.
  • 日本学術振興会：科学研究費助成事業
    Date (from‐to) : 1991 -1991 
    Author : 奥山 哲郎; 河田 成人; 加戸 次郎; 住岡 武; 津島 行男; 原田 学
     
    本研究課題では、環の表現論と有限群のコホモロジィ理論における最近の進展を見すえ、有限群の表現論との関わり、互いの応用を研究することを目標とした。 研究実施計画(1)、(2)にそって、1.群環上のAuslanderーReiten図の形状の分類に取り組んだ。この形状の分類は群環の加群の研究の中心課題のひとつで多くの情報を与えるものである。(1).河田らを中心に、次元にある条件を与えて形状の分類を得た。研究方法はAuslanderーRecten列の構成にも応用でき興味深く今後さらに進展させたい。(2).奥山らを中心に、vertexが2種類以上の場合に分類を得た。主道具はgreen対応とclifford理論を発展させたもので新しい方法である。2.環の表現論における相対射影性,移入性の理論を発展させた。原田らを中心に、ある環の特性をこの概念を用いて特徴付け、新しい視点を加えた。群の表現論への応用の可能性をもち、ひきつづき研究を進めたい。 研究実施計画(3),(4)に関連して、群のコホモロジィ理論への表現論の応用に取り組んだ。津島,奥山らを中心に実行されたが,1.加群の相対射影被覆について新しい等式を得ることができた。この等式は計算を実行する際きわめて有効で広い応用をもつものである。2.実際にその応用として、ある種のクラスの有限群のコホモロジィ環の計算を実行した。特に、extraーspecial pー群の自明な加群の相対射影視覆について興味ある事実を発見した。この麦は一般の有限群のコホモロジィ環を決定する上で重要な位置を占めるもので,Carlsonの加群の指数についての問題を解決するための重要な一歩を与えると考える。 研究分担者間の協力は十分にでき,経費の多くをあてておこなった全国各地の研究者との交流・討論も有意義であった。研究課題の遂行にあたっての経費の補助に感謝します。

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