Lower defect groups and vertices of simple modules Akihiko Hida; Masao Kiyota
Journal of Algebra,
Volume：617,
First page：113,
Last page：126, Mar. 2023,
[Reviewed]Elsevier BV, Scientific journal
DOI：https://doi.org/10.1016/j.jalgebra.2022.10.037DOI ID：10.1016/j.jalgebra.2022.10.037,
ISSN：0021-8693 The splitting of cohomology of pgroups with rank 2 Akihiko Hida; Nobuaki Yagita
Forum Mathematicum,
Volume：30,
Number：1,
First page：191,
Last page：212, Jan. 2018,
[Reviewed]Let p be an odd prime and BP the classifying space of a p-group P with rankp(P) = 2. Using stable homotopy splitting of BP, we study the decomposition of cohomology Heven(P
Z)/p.
Walter de Gruyter GmbH, English, Scientific journal
DOI：https://doi.org/10.1515/forum-2016-0241DOI ID：10.1515/forum-2016-0241,
ISSN：1435-5337,
SCOPUS ID：85040176377 Representations of the double Burnside algebra and cohomology of the extraspecial p-group II Akihiko Hida; Nobuaki Yagita
JOURNAL OF ALGEBRA,
Volume：451,
First page：461,
Last page：493, Apr. 2016,
[Reviewed]Let E be the extraspecial p-group of order p(3) and exponent p where p is an odd prime. We determine the mod p cohomology H*(X,F-p) of a summand X in the stable splitting of p -completed classifying space BE. In the previous paper (Hida and Yagita (2014) [7]), we determined this cohomology modulo nilpotence. In this paper, we consider the whole of the cohomology. Moreover, we consider the stable splittings of BG for some finite groups with Sylow p-subgroup E related to the three dimensional linear group L-3 (p). (C) 2015 Elsevier Inc. All rights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
DOI：https://doi.org/10.1016/j.jalgebra.2015.12.003DOI ID：10.1016/j.jalgebra.2015.12.003,
ISSN：0021-8693,
eISSN：1090-266X,
Web of Science ID：WOS:000370311700016 Representations of the double Burnside algebra and cohomology of the extraspecial p-group Akihiko Hida; Nobuaki Yagita
JOURNAL OF ALGEBRA,
Volume：409,
First page：265,
Last page：319, Jul. 2014,
[Reviewed]Let E be the extraspecial p-group of order p(3) and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors. (C) 2014 Elsevier Inc. All rights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
DOI：https://doi.org/10.1016/j.jalgebra.2014.03.021DOI ID：10.1016/j.jalgebra.2014.03.021,
ISSN：0021-8693,
eISSN：1090-266X,
Web of Science ID：WOS:000349811300012 Module Correspondences in Rouquier Blocks of Finite General Linear Groups
Akihiko Hida; Hyohe Miyachi
REPRESENTATION THEORY OF ALGEBRAIC GROUPS AND QUANTUM GROUPS, Volume：284, First page：81, Last page：92, 2010, [Reviewed]
In this chapter we shall consider l-modular represenations of finite general linear groups in non-defining characteristic l > 0. We focus the nicest l-block algebras in this representation theory, which are also known as unipotent Rouquier blocks.
Let B(w,rho) be the unipotent l-block algebra of a general linear group over the finite field with q elements associated with an e-weight w > 0 and a Rouquier e-core rho with respect to w where e is the multiplicative order of q modulo l > 0. (See the second paragraph of Sect. A for the definition of rho.)
We assume that B(w,p) has an abelian defect, ie, l > w. It is known that there exists a Morita equivalence F between B(w,rho) and the wreath product block B1,(sic). This result is obtained by W. Turner and the second author independently. The both methods are completely identical each other and are very similar to J. Chuang and R. Kessar's method on symmetric groups.
R. Kessar's method on symmetric groups.
In this chapter, under the equivalence F we shall determine the explicit correspondences of the simple, the Young and the Specht modules over the Rouquier block B(w,p) and the local subgroup block B(sic). The result is used to prove the intial condition of runner removal Morita equivalence theorem.
BIRKHAUSER VERLAG AG, English, International conference proceedings
ISSN：0743-1643, Web of Science ID：WOS:000297616400005
Control of fusion and cohomology of trivial source modules Akihiko Hida
JOURNAL OF ALGEBRA,
Volume：317,
Number：2,
First page：462,
Last page：470, Nov. 2007,
[Reviewed]Let G be a finite group and H a subgroup. We give an algebraic proof of Mislin's theorem which states that the restriction map from G to H on mod-p cohomology is an isomorphism if and only if H controls p-fusion in G. We follow the approach of P. Symonds [P. Symonds, Mackey functors and control of fusion, Bull. London Math. Soc. 36 (2004) 623-632] and consider the cohomology of trivial source modules. (c) 2007 Elsevier Inc. All rights reserved.
ACADEMIC PRESS INC ELSEVIER SCIENCE, English, Scientific journal
DOI：https://doi.org/10.1016/j.jalgebra.2007.08.008DOI ID：10.1016/j.jalgebra.2007.08.008,
ISSN：0021-8693,
Web of Science ID：WOS:000251682200002 Morita equivalent blocks in non-normal subgroups and p-radical blocks in finite groups
A Hida; S Koshitani
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, Volume：59, Number：2, First page：541, Last page：556, Apr. 1999, [Reviewed]
Let O be a complete discrete valuation ring with unique maximal ideal J(O), let K be its quotient field of characteristic 0, and let k be its residue field O/J(O) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the [GI-th roots of unity, where /G/ is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose: that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R is an element of{O,k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with 'naturally Morita equivalent blocks A and B', which means that A is isomorphic to a full matrix algebra of B, as studied by B. Kulshammer. However, Kulshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Kulshammer. Moreover, in the case His normal in G, we get the same results as Kulshammer; however, he uses the results of E. C. Dade, and we do not.
OXFORD UNIV PRESS, English, Scientific journal
ISSN：0024-6107, Web of Science ID：WOS:000082452100012
Some remarks on the loewy series of projective modules for p-solvable groups
A Hida
COMMUNICATIONS IN ALGEBRA, Volume：25, Number：12, First page：3713, Last page：3719, 1997, [Reviewed]
TAYLOR & FRANCIS INC, English, Scientific journal
ISSN：0092-7872, eISSN：1532-4125, Web of Science ID：WOS:A1997YK04400001
A NOTE ON KERNELS AND VERTICES OF SIMPLE MODULES
A HIDA
JOURNAL OF ALGEBRA, Volume：171, Number：3, First page：917, Last page：920, Feb. 1995, [Reviewed]
ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS, English
ISSN：0021-8693, Web of Science ID：WOS:A1995QJ14800011
ON P-RADICAL BLOCKS OF FINITE-GROUPS
A HIDA
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Volume：114, Number：1, First page：37, Last page：38, Jan. 1992, [Reviewed]
We give a sufficient condition on a p-block of a finite group under which the block is p-radical.
AMER MATHEMATICAL SOC, English, Scientific journal
ISSN：0002-9939, Web of Science ID：WOS:A1992HA42800006
Cohomology of the Extraspecial $p$-Group and Representations of the Double Burnside Algebra
Hida Akihiko
RIMS Kokyuroku, Volume：1872, First page：132, Last page：139, Jan. 2014
Kyoto University, Japanese
ISSN：1880-2818, CiNii Articles ID：110009676099, CiNii Books ID：AN00061013
Mackey functor and cohomology of finite groups
A. Hida
Volume：1581, First page：1, Last page：5, 2008
有限群の表現論とコホモロジーに関する研究
飛田明彦
Volume：第5号(18年度), First page：392, Last page：393, 2007
対称群のブルエ予想
飛田明彦
First page：137, Last page：150, 2007
Mackey functor and cohomology of finite groups
飛田明彦
First page：124, Last page：127, 2007
Control of fusion and cohomology of finite groups
飛田明彦
Volume：1466, First page：55, Last page：60, 2006
Extensions and Cohomology of Association Schemes (Algebraic Combinatorics)
Hida Akihiko
RIMS Kokyuroku, Volume：1394, First page：47, Last page：51, 2004
Kyoto University, Japanese
ISSN：1880-2818, CiNii Articles ID：120000903756, CiNii Books ID：AN00061013
ON THE PRINCIPAL BLOCKS OF FINITE GENERAL LINEAR GROUPS IN NON-DEFINING CHARACTERISTIC
Hida Akihiko; Miyachi Hyoue
RIMS Kokyuroku, Volume：1140, First page：127, Last page：130, Apr. 2000
Kyoto University, English
ISSN：1880-2818, CiNii Articles ID：110000164257, CiNii Books ID：AN00061013
Cohomology of finite groups and homotopy theory of classifying spaces from the viewpoint of representation theory
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 01 Apr. 2021 - 31 Mar. 2026
Saitama University
Grant amount(Total)：2340000, Direct funding：1800000, Indirect funding：540000
Grant number：21K03154
Cohomology theory of finite groups from the viewpoint of representation theory
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 01 Apr. 2016 - 31 Mar. 2020
Hida Akihiko, Saitama University
Grant amount(Total)：2080000, Direct funding：1600000, Indirect funding：480000
In this research, we study the relation among the following three objects on finite groups, stable splitting of classifying space, p-local structure for a prime p, modular representations. We consider rank two p-groups, in particular, we study extra special p-groups. We determine the stable irreducible summands of stable splitting and the cohomology of the summand. We use the representations of double Burnside algebra and the theory of biset functors .
Grant number：16K05054
Cohomology of finite groups and homotopy theory of classifying space from the view point of representation theory
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 01 Apr. 2012 - 31 Mar. 2015
HIDA Akihiko, Saitama University
Grant amount(Total)：1430000, Direct funding：1100000, Indirect funding：330000
The double Burnside algebra of a finite group is the algebra with basis corresponding to finite sets such that the group acts from both sides. This algebra contains the information on the splitting of classifying space of finite group in the stable homotopy category. We studied the stable splitting of classifying space of finite groups through the action of double Burnside algebra on the cohomology.
We mainly consider the nonabelian p-group of order p cubed and exponent p. We determined the multiplicity of summands in the stable splitting and cohomology of these summands with coefficient the field of p elements. Moreover, we obtain information on finite groups having this group as a Sylow p-subgroup.
Grant number：24540007
Generalizations of perfect isometries between the sets of characters of finite groups
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 01 Apr. 2010 - 31 Mar. 2015
UNO KATSUHIRO; WAKI Katsushi; KUNUGI Naoko; HIDA Akihiko; NARASAKI Ryo; ARIKI Susumu; MIYACHI Hyoue; OKUYAMA Tetsuro
Grant amount(Total)：3380000, Direct funding：2600000, Indirect funding：780000
We classify finite simple groups having small Sylow subgroups and determine the structure of groups appearing as Sylow subgroups of those finite simple groups. Moreover, in the case where the prime is small and a Sylow group is not abelian, we determine fusion systems over such Sylow groups.
For finite simple groups appearing in the above classification with the same Sylow subgroups and the same fusion systems over them, we construct perfect isometries between the set of irreducible characters belonging to the principal blocks of them. These isometries are those defined by Broue, namely, the ordinary perfect isometries, and not generalizations of them.
Grant number：22540021
Cohomology theory of finite groups
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2010 - 2012
SASAKI Hiroki; WATANABE Atumi; OKUYAMA Tetsuro; HIDA Akihiko, Shinshu University
Grant amount(Total)：2340000, Direct funding：1800000, Indirect funding：540000
Block ideals are main objects to be investigated in representation theory of finite groups. In this study we gave a characterization of cohomology rings of block ideal in terms of source algebras of block ideals. Source algebras are so important invariants of block ideals. However it is so hard to treat with them; we gave a cohomological criterion for analyzing source algebras. Applying these results to block ideals of tame representation type, we investigated cohomology rings and source algebras.
Grant number：22540013
Analysis of decomposition matrices for sporadic finite simple groups
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2010 - 2012
WAKI Katsushi; HIDA Akihiko; HANAKI Akihide; KUNUGI Naoko, Yamagata University
Grant amount(Total)：4030000, Direct funding：3100000, Indirect funding：930000
It has been clear that the calculations of characters in the sporadic finite simple group J4 which is the main object in this project and its maximal subgroup are not enough for the decision of decomposition matrix of the full defect blocks in J4. We found the concrete way of the construction of modular representations over a field of characteristic 3 by the algebra analysis system GAP using the Amagamation from two representations of maximal subgroups of J4. This construction is deduced from the construction of the ordinary representation of J4. We also proved the irreducible p-module of dimension 1333 of J4 is trivial source module in case that p=3.
Grant number：22540007
Cohomology of finite groups from the view point of representation theory
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2009 - 2011
HIDA Akihiko, Saitama University
Grant amount(Total)：1170000, Direct funding：900000, Indirect funding：270000
We studied the cohomology theory of finite groups and finite dimensional algebras using methods of representation theory. We considered the varieties of modules defined by Hochschild cohomology and the rank varieties of modules over exterior algebras or graded Hopf algebras. In particular, we obtained some results on tensor products of modules. On the other hand, we studied the action of double Burnside algebra on the mod-p cohomology algebra of a finite group and obtained some results on composition factors.
Grant number：21540007
Structure of the derived categories of block algebras with non-commutative defect groups
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), 2006 - 2008
UNO Katsuhiro; ASASHIBA Hideto; WAKI Katsushi; KUNUGI Naoko; HIDA Akihiko; BABA Yoshitomo; HIRAKI Akira; HIRAKI Akira, Osaka Kyoiku University
Grant amount(Total)：4150000, Direct funding：3400000, Indirect funding：750000
Grant number：18540031
Applications of Cohomology groups and Shintani descent to Broue's conjecture
Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), 2002 - 2005
UNO Katsuhiro; KAWANAKA Noriaki; HIRAKI Akira; WAKI Katsushi; HIDA Akihiko; KUNUGI Naoko
Grant amount(Total)：10000000, Direct funding：10000000
1 (1)For low rank finite Chevalley groups G of type D and F, and for p=5, we constructed a perfect isometry between the principal blocks of G and a Sylow normalizer. We also determine the number of isometries. Moreover, for blocks of cyclic defect groups, we proved that for any bijection between irreducible characters there is a derived equivalence inducing the bijecyion, provided that the local structure of the blocks are the same,
(2)For finite Chevaley groups of type G, and for p=3, we establish certain Morita equivalence between principal blocks for several defining fields.
(3)We proved that Broue's conjecture holds for block with elementary abelian defect groups of order 9.
(4)We have obtained an easy formula for elementary divisors of the Cartan matrices of symmetric groups.
(5)We gave a module theoretic proof for the Mislin's theorem on group fusions.
(6)The fundamental notions of group cohomologies are extended for arbitrary blocks.
2 (1)We extend Dade's conjecture, and check that it holds for many sporadic simple groups and some finte Chevaley groups.
(2)For a block with dihedral defect groups of order eight, we construct functors among the derived categories of normalizers of radical chains which induce the relations between the numbers of simple modules.
Grant number：14340012
アウスランダーライテン理論における既約加群の役割について
1999 - 2000
Grant amount(Total)：2200000, Direct funding：2200000
Grant number：11874006
有限次元多元環の研究
1996 - 1996
Grant amount(Total)：2000000, Direct funding：2000000
Grant number：08640012
距離空間における次元の研究
1995 - 1995
Grant number：07640094
-
Competitive research funding
