越谷重夫

コシタニシゲオ (Shigeo Koshitani)

基本情報

所属: 千葉大学先進科学センター特任教授 (千葉大学理学研究科名誉教授)

学位: アーベル2一シロー部分群をもつ有限群の主2一プロックについて(1980年7月筑波大学)

J-GLOBAL ID: 200901010238520987
researchmap会員ID: 1000010762

外部リンク: http://www.math.s.chiba-u.ac.jp/~koshitan/index.html

研究キーワード

研究分野

自然科学一般 / 代数学 /

委員歴

2005年7月 - 2007年6月

日本数学会日本数学会雑誌「数学」編集委員長

論文

Recognition of Brauer indecomposability for a Scott module

Shigeo Koshitani, İpek Tuvay

Journal of Algebra and Its Applications 2023年9月7日
The Brauer indecomposability of Scott modules with wreathed 2-group vertices

Shigeo Koshitani, Ipek Tuvay

Rocky Mountain Journal of Mathematics 2022年1月査読有り筆頭著者最終著者
Brauer indecomposability of Scott modules with semidihedral vertex

Shigeo Koshitani, İpek Tuvay

Proceedings of the Edinburgh Mathematical Society 1-9 2021年5月4日査読有り筆頭著者最終著者

<title>Abstract</title> We present a sufficient condition for the <inline-formula> <alternatives> <tex-math>$kG$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline1.png" /> </alternatives> </inline-formula>-Scott module with vertex <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline2.png" /> </alternatives> </inline-formula> to remain indecomposable under the Brauer construction for any subgroup <inline-formula> <alternatives> <tex-math>$Q$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline3.png" /> </alternatives> </inline-formula> of <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline4.png" /> </alternatives> </inline-formula> as <inline-formula> <alternatives> <tex-math>$k[Q\,C_G(Q)]$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline5.png" /> </alternatives> </inline-formula>-module, where <inline-formula> <alternatives> <tex-math>$k$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline6.png" /> </alternatives> </inline-formula> is a field of characteristic <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline7.png" /> </alternatives> </inline-formula>, and <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline8.png" /> </alternatives> </inline-formula> is a semidihedral <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline9.png" /> </alternatives> </inline-formula>-subgroup of a finite group <inline-formula> <alternatives> <tex-math>$G$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline10.png" /> </alternatives> </inline-formula>. This generalizes results for the cases where <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline11.png" /> </alternatives> </inline-formula> is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline12.png" /> </alternatives> </inline-formula>-permutation bimodule (where <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline13.png" /> </alternatives> </inline-formula> is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.
Trivial source characters in blocks with cyclic defect groups

Shigeo Koshitani, Caroline Lassueur

Journal of Algebra 574 375-408 2021年5月査読有り筆頭著者
Splendid Morita equivalences for principal blocks with semidihedral defect groups

Shigeo Koshitani, Caroline Lassueur, Benjamin Sambale

Proceedings of the American Mathematical Society 1-1 2021年4月23日査読有り筆頭著者最終著者

もっとみる

MISC

指数$p$の巡回部分群を持つ群が不足群であるブロック (有限群のコホモロジー論の研究)

Koshitani Shigeo

数理解析研究所講究録 1581 20-22 2008年2月
Broue's abelian defect group conjecture holds for the Janko simple group J_4

KOSHITANI S.

J. Pure Appl. Algebra 212 1438-1456 2008年
The Isaacs corresponding blocks of finite groups (群論とその周辺--RIMS研究集会報告集)

越谷重夫

数理解析研究所講究録 1564 58-60 2007年7月
An extension of Watanabe's theorem for the Isaacs-Horimoto-Watanabe corresponding blocks

ME Harris, S Koshitani

JOURNAL OF ALGEBRA 296(1) 96-109 2006年2月
The indecomposability of a certain bimodule given by the Brauer construction

S Koshitani, M Linckelmann

JOURNAL OF ALGEBRA 285(2) 726-729 2005年3月

Broue's abelian defect conjecture [Asterisque 181/182 (1990) 61-92. 6.2] predicts for a p-block of a finite group G with an abelian defect group P a derived equivalence between the block algebra and its Brauer correspondent. By a result of Rickard [J. London Math. Soc. 43 (1991) 37-48], such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule-and it appears that these stable equivalences in turn tend to be obtained by "gluing" together Morita equivalences at the local levels of the considered blocks see. e.g., [M. Broue, Equivalences of blocks of group algebras, in: V Dlab, L.L. Scott (Eds.). Finite Dimensional Algebras and Related Topics, Kluwer Acad. Publ., 1994, pp. 1-26, 6.3], [M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001) 819-843, 3.1] [J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996) 331-358, 4.1], and [R. Rouquier, Block theory via stable and Rickard equivalences, in: M.J. Collins, B.J. Parshall, L.L. Scott (Eds.), Modular Representation Theory of Finite Groups., de Gruyter, Berlin, 2001, pp. 101-146, 5.6, A.4.1]. This note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above. This is used in [S. Koshitani, N. Kunugi, K. Waki, Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004)638-666] to show that Broue's abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group. (c) 2004 Elsevier Inc. All rights reserved.
The indecomposability of a certain bimodule given by the Brauer construction

S Koshitani, M Linckelmann

JOURNAL OF ALGEBRA 285(2) 726-729 2005年3月

Broue's abelian defect conjecture [Asterisque 181/182 (1990) 61-92. 6.2] predicts for a p-block of a finite group G with an abelian defect group P a derived equivalence between the block algebra and its Brauer correspondent. By a result of Rickard [J. London Math. Soc. 43 (1991) 37-48], such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule-and it appears that these stable equivalences in turn tend to be obtained by "gluing" together Morita equivalences at the local levels of the considered blocks see. e.g., [M. Broue, Equivalences of blocks of group algebras, in: V Dlab, L.L. Scott (Eds.). Finite Dimensional Algebras and Related Topics, Kluwer Acad. Publ., 1994, pp. 1-26, 6.3], [M. Linckelmann, On splendid derived and stable equivalences between blocks of finite groups, J. Algebra 242 (2001) 819-843, 3.1] [J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996) 331-358, 4.1], and [R. Rouquier, Block theory via stable and Rickard equivalences, in: M.J. Collins, B.J. Parshall, L.L. Scott (Eds.), Modular Representation Theory of Finite Groups., de Gruyter, Berlin, 2001, pp. 101-146, 5.6, A.4.1]. This note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above. This is used in [S. Koshitani, N. Kunugi, K. Waki, Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group, J. Algebra 279 (2004)638-666] to show that Broue's abelian defect group conjecture holds for nonprincipal blocks of the simple Held group and the sporadic Suzuki group. (c) 2004 Elsevier Inc. All rights reserved.
Blocks of central p-group extensions（共著）

Proceedings of the American Mathematical Society 133(1) 21-26 2005年1月
BLOCKS OF CENTRAL p-GROUP EXTENSIONS

Shigeo Koshitani, Naoko Kunugi

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 133(1) 21-26 2005年

Let G and G' be finite groups that have a common central p-subgroup Z for a prime number p, and let (A) over bar and (A') over bar respectively be p-blocks of G/Z and G'/Z induced by p-blocks A and A' respectively of G and G', both of which have the same defect group. We prove that if (A) over bar and (A') over bar are Morita equivalent via a certain special ((A) over bar, (A') over bar)-bimodule, then such a Morita equivalence lifts to a Morita equivalence between A and A'.
Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group

S Koshitani, N Kunugi, K Waki

JOURNAL OF ALGEBRA 279(2) 638-666 2004年9月

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broue. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding p-block B of N-G (P) are derived equivalent. We demonstrate in this paper that Broue's conjecture holds for non-principal 3-blocks A with elementary abelian defect group P of order 9 of the simple Held group and the sporadic simple Suzuki group. (C) 2004 Elsevier Inc. All rights reserved.
Broue's abelian defect group conjecture for the Held group and the sporadic Suzuki group(共著)

Journal of Algebra 279(2) 638-666 2004年9月
Conjectures of Donovan and Puig for principal 3-blocks with abelian defect groups

S Koshitani

COMMUNICATIONS IN ALGEBRA 31(5) 2229-2243 2003年

One of the most important and interesting conjectures in representation theory of finite groups is Donovan's conjecture. It is on block algebras of group algebras of finite groups over an algebraically closed field of prime characteristic p. Donovan conjectures that, for any given finite p-group D, up to Morita equivalence, there are only finitely many block algebras with defect group D. We prove in this article that Donovan's conjecture is true for principal block algebras in the case where D is an arbitrary finite abelian 3-group. There is another important and interesting conjecture due to Puig, which is stated as the above conjecture if we replace "Morita equivalence" by "Puig equivalence". We prove in this paper that Puig's conjecture is true for principal 3-block algebras of finite groups with an elementary abelian Sylow 3-subgroup of order 9. Actually, we prove even more. It is proven in this paper also that there are exactly 22 non-Puig equivalent classes of principal block algebras of finite groups over a complete discrete valuation king of rank one if a Sylow 3-subgroup D of them is an elementary abelian 3-group of order 9. The results here depend on the classification of finite simple groups.
Broue's conjecture for non-principal 3-blocks of finite groups(共著)

Journal of Pure and Applied Algebra 173(2) 177-211 2002年8月24日
Broue's conjecture for non-principal 3-blocks of finite groups(共著)

Journal of Pure and Applied Algebra 173(2) 177-211 2002年8月
Broue's conjecture holds for principal 3-blocks with elementary abelian defect group of order 9(共著)

Journal of Algebra 248(2) 575-604 2002年2月
Broue's conjecture holds for principal 3-blocks with elementary abelian defect group of order 9(共著)

Journal of Algebra 248(2) 575-604 2002年2月
Glauberman correspondence of p-blocks of finite groups(共著)

Journal of Algebra 243(2) 504-517 2001年9月
Glauberman correspondence of p-blocks of finite groups(共著)

Journal of Algebra 243(2) 504-517 2001年9月
The principal 3-blocks of the 3-dimensional projective special unitary groups in non-defining characteristic(共著)

Journal fuer reine und angewandte Mathematik 539, 1-27 2001年
Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian sylow 3-group of order 9(共著)

Communications in Algebra 29(10) 4509-4522 2001年
Donovan conjecture and Loewy length for principal 3-blocks of finite groups with elementary abelian sylow 3-group of order 9(共著)

Communications in Algebra 29, 4509-4522 2001年
A Remark of the Loewy Structure for the Three Dimensional Projective Special Unitary Groups in Characteristic 3(共著)

Advanced Studies in Pure Mathematics 32, 337-346 2001年
The Principal 3-Blocks of Four-and Five-Dimensional Projective Special Linear Groups in Non-Defining Characteristic(共著)

Journal of Algebra 226(2) 788-806 2000年4月15日
The Principal 3-Blocks of Four-and Five-Dimensional Projective Special Linear Groups in Non-Defining Characteristic(共著)

Journal of Algebra 226(2) 788-806 2000年4月
A remark on $p$-blocks of finite groups with abelian defect groups (Representation Theory of Finite Groups and Related Topics)

越谷重夫

数理解析研究所講究録 1149 34-38 2000年4月
Morita Equivalent Blocks in Non-normal Subgroups and p-Radical Blocks in Finite Groups(共著)

Journal of the London Mathematical Society 59(59) 541-556 1999年4月
The Green Correspondents of the Mathieu Group M12 in Characteristic 3(共著)

Communications in Algebra 27(1) 37-66 1999年
The Loewy Structure of the Projective Modules of the Mathier Group M12 and Its Automorphism Group in Characteristic 3(共著)

Communications in Algebra (27) 1-36 1999年
Morita Equivalent Blocks in Non-normal Subgroups and p-Radical Blocks in Finite Groups(共著)

Journal of the London Mathematical Society (59) 541-556 1999年
The Loewy Structure of the Projective Modules of the Mathier Group M12 and Its Automorphism Group in Characteristic 3(共著)

Communications in Algebra (27) 1-36 1999年
Local Subgrops and Group Algebras of Finite P-Solvable Groups(共著)

Algebras and Representation Theory 1(2) 155-159 1998年
Cartan invariants of group algebras of finite groups

S Koshitani

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 124(8) 2319-2323 1996年8月

We give a result on Cartan invariants of the group algebra kG of a finite group G over an algebraically closed field k, which implies that if the Loewy length (socle length) of the projective indecomposable kG-module corresponding to the trivial kG-module is four, then k has characteristic 2. The proof is independent of the classification of finite simple groups.
A Splitting Theorem for Blocks(共著)

Osaka Journal of Mathematics 33(2) 343-346 1996年6月
A Splitting Theorem for Blocks(共著)

Osaka Journal of Mathematics 33(2) 343-346 1996年6月
Cartan invariants of group algebras of finite groups

Shigeo Koshitani

Proceedings of the American Mathematical Society 124(8) 2319-2323 1996年

We give a result on Cartan invariants of the group algebra kG of a finite group G over an algebraically closed field k, which implies that if the Loewy length (socle length) of the projective indecomposable kG-module corresponding to the trivial kG-module is four, then k has characteristic 2. The proof is independent of the classification of finite simple groups. © 1996 American Mathematical Society.
A Quiver and Relations for Some Group Algebras of Finite Groups(Representation Theory of Finite Groups and Algebras)

越谷重夫

数理解析研究所講究録 877 50-51 1994年6月
Projective Modules of Finite Groups with Elementary Abelian Sylow-3-Subgroups of Order 9 in Characteristic 3

Proceedings of the Royal Society of Edinburgh 124A 161-168 1994年
PROJECTIVE-MODULES OF FINITE-GROUPS WITH ELEMENTARY ABELIAN SYLOW-3-SUBGROUPS OF ORDER-9 IN CHARACTERISTIC-3

S KOSHITANI

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 124 161-168 1994年

Let G be any finite group with elementary abelian Sylow 3-subgroups of order 9, and let F be any field of characteristic 3. Then, the Loewy length of the projective cover of the trivial FG-module is at least 5. This lower bound is the best possible.
On p-Radical Groups in Representation Theory of Finite Groups

Canadian Mathematical Society Proceedings 11, 209-212 1991年
On p-Radical Groups in Representation Theory of Finite Groups

Canadian Mathematical Society Proceedings 11, 209-212 1991年
A REMARK ON P-RADICAL GROUPS

S KOSHITANI

JOURNAL OF ALGEBRA 134(2) 491-496 1990年11月
ON THE KERNELS OF REPRESENTATIONS OF FINITE-GROUPS .2.

S KOSHITANI

GLASGOW MATHEMATICAL JOURNAL 32 341-347 1990年9月
A Remark on p-Radical Groups

Journal of Algebra 134, 491-496 1990年
On the kernels of representations of finite groups II

Shigeo Koshitani

Glasgow Mathematical Journal 32(3) 341-347 1990年
The Nilpotency Index of the radicals of Group ALgebras of Finite Groups WHose Sylow 3-Subgroups are Extra-Special of order 27 of index 3

Proceedings of the Royal Society of Edinburgh 108A, 117-132 1988年
THE NILPOTENCY INDEX OF THE RADICALS OF GROUP-ALGEBRAS OF FINITE-GROUPS WHOSE SYLOW 3-SUBGROUPS ARE EXTRA-SPECIAL OF ORDER-27 OF EXPONENT-3

S KOSHITANI

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS 108 117-132 1988年
The loewy structure of the projective indecomposable modules for SL(3,3) and its automorphism group in characteristic 3

Shigeo Koshitani

Communications in Algebra 15(6) 1215-1253 1987年1月1日
THE LOEWY STRUCTURE OF THE PROJECTIVE INDECOMPOSABLE MODULES FOR SL(3,3) AND ITS AUTOMORPHISM GROUP IN CHARACTERISTIC-3

S KOSHITANI

COMMUNICATIONS IN ALGEBRA 15(6) 1215-1253 1987年
ON GROUP-ALGEBRAS OF FINITE-GROUPS

S KOSHITANI

LECTURE NOTES IN MATHEMATICS 1178 109-128 1986年
A REMARK ON THE JACOBSON RADICAL OF A BLOCK IDEAL IN A FINITE P-SOLVABLE GROUP

S KOSHITANI

JOURNAL OF ALGEBRA 94(1) 106-112 1985年
ON THE LOEWY SERIES OF THE GROUP-ALGEBRA OF A FINITE P-SOLVABLE GROUP WITH P-LENGTH-GREATER-THAN-1

S KOSHITANI

COMMUNICATIONS IN ALGEBRA 13(10) 2175-2198 1985年

講演・口頭発表等

指標が少ない主ブロック

越谷重夫

表現論の日々2019, ドイツ・イェーナ大学 2019年10月26日招待有り
ブロック同値とブロック不変量

越谷重夫

ブロック理論の上級セミナー,ドイツ・カイザースラウターン工科大学 2019年7月8日招待有り
小さなブロック

越谷重夫

表現論の日々2018, ドイツ, ハノーファー 2018年9月14日招待有り
主ブロックの素晴らしい森田同値

越谷重夫

理科大表現論セミナー 2018年2月17日招待有り
アウスランダー・ライテン箙における有限群代数の単純加群

越谷重夫

ドイツ・ブッパルタール大学代数・数論セミナー 2017年11月14日招待有り

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所属学協会

日本数学会

Works(作品等)

Infinite Representation Type for Group Algebras of Finite Groups

1973年 - 2017年
有限群の群多元環上の無限型表現

1973年 - 2017年
Block Theory of Finite Groups

1999年 - 2006年
有限群のブロック理論

1999年 - 2006年

共同研究・競争的資金等の研究課題

有限群の表現論における局所大域予想

日本学術振興会科学研究費助成事業 2019年4月 - 2023年3月

越谷重夫
有限群の表現論におけるブルエ予想およびその周辺

日本学術振興会科学研究費助成事業 2015年4月 - 2018年3月

越谷重夫
有限群の表現論におけるブルエ予想の解決

日本学術振興会科学研究費助成事業 2011年4月 - 2015年3月

越谷重夫, 功刀直子
環と加群圏の代数構造の表現論的研究

日本学術振興会科学研究費助成事業 2009年 - 2011年

山形邦夫, 伊山修, 越谷重夫, 浅芝秀人, 大城紀代一, 佐藤眞久, 西田憲司, 毛利出, 吉野雄二, 和田幸
有限群の表現論におけるブルエ予想の解決

日本学術振興会科学研究費助成事業 2008年 - 2010年

越谷重夫, 野澤宗平, 北詰正顕, 西田康二, 松田茂樹, 大坪紀之, 功刀直子

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越谷 重夫

基本情報

研究キーワード

研究分野

委員歴

論文

MISC

講演・口頭発表等

所属学協会

Works(作品等)

共同研究・競争的資金等の研究課題

越谷重夫