RESEARCH PROJECT
Equivalences of Categories in the Representation Theory of Finite Groups
Versiunea in limba romana
Information
Program: PN2 IDEI (Informations on the program at CNCSIS)
Project type: PN-II-ID-PCE-2007-1 (Exploratory Research Projects)
Contracting authority: UEFISCSU
Contractor: Babes-Bolyai University Cluj-Napoca
Project code: ID_532
Contract no: 29/28.09.2007
Duration: 3 years, 01.10.2007-30.09.2010
Research team
Prof. dr. Andrei Marcus, project director
Assistant prof. Camelia Dicu, researcher
Assistant prof. Gabriela Olteanu, researcher
PhD student Constantin Cosmin Todea, research assistant
PhD student Tiberiu Coconet, research assistant
Abstract
The topics of this project are finite groups and their modular and projective representations; derived equivalences between blocks of group algebras; blocks of normal subgroups, Clifford theory; nilpotent blocks, hyperfocal subalgebra of a block. The research in this field is very intense on an international level, and has multiple ramifications and applications. We intend to combine classical methods with new methods. The general aim to attack modern subjects in order to integrate in the international scientific community and the formation of young researchers. Main directions of investigation (of theoretical and computational nature): a categorical point of view on Clifford theoretical invariants related to conjectures of Alperin-Dade-Isaacs-Navarro-Uno, equivalences of Brauer categories, invariance of the Harris-Knoerr correspondence under derived equivalence, the hyperfocal subalgebra of block extensions, obtaining Rickard equivalences from perfect isometries and isotypies of characters, reduction for Donovan's conjecture to quasisimple groups, computation of the block cohomology, and of Schur indices.
Activities
  • Research seminar: weekly, Wednesday 10:00-12:00, UBB, building Mathematicum, room pi.
  • Research visit: Andrei Marcus, 13-20.10.2007, prof. dr. Stefaan Caenepeel, Vrije Universiteit Brussel (VUB); Algebra seminar talk Crossed products, Brauer groups and Clifford classes on 19.10.2007.
  • Research visit: Andrei Marcus, 9-16.11.2007, prof. dr. Thorsten Holm, Otto-von-Guericke-Universität Magdeburg and Leibniz Universität Hannover; Algebra and Algebraic Combinatorics seminar talk Derived invariance of Clifford classes, Hannover on 13.11.2007.
  • Public defence of Ph.D. thesis: Gabriela Olteanu, 19.11.2007, title: Wedderburn Decomposition of Group Algebras. A Computational Approach with Applications to Schur Groups and Units, Universidad de Murcia, Spain, supervisors: prof. dr. Ángel del Río and prof. dr. Juan Jacobo Simon;
  • Research visit: Gabriela Olteanu, 13.01-17.02.2008, prof. dr. Ángel del Río, Universidad de Murcia, Spain; introduction to the theory of association schemes.
  • Public defence of Ph.D. thesis: Camelia Dicu, 7.03.2008, title: Metode modul-teoretice in studiul G-algebrelor si al grupurilor punctate, Babes-Bolyai University, Cluj-Napoca, supervisor: prof. dr. Andrei Marcus.
  • Research visit: Tiberiu Coconet, 02.03-08.03.2008, Universität Wuppertal, Germany.
  • Research visit: Andrei Marcus, 27.03-10.04.2008; participation at the program Representation Theory of Finite Groups and Related Topics, ordinary member, Mathematical Sciences Research Institute, Berkeley, USA.
  • Research visit: Tiberiu Coconet, Cosmin Todea, 30.04-04.05.2008, Universität Bielefeld, Germany.
  • Organization of conference: International Conference on "Modules and Representation Theory", Babes-Bolyai University Cluj-Napoca, July 7-12, 2008. Talk given by Gabriela Olteanu with the title Ring isomorphism of cyclic cyclotomic algberas, 08.07.2008.
  • Research visit: Tiberiu Coconet, Andrei Marcus, Gabriela Olteanu, Cosmin Todea, 18-24.08.2008, Dr. John Murray, National University of Ireland, Maynooth, Ireland. Algebra seminar talk given by Andrei Marcus, with the title Derived invariance of Clifford classes, 19.08.2008, and by Gabriela Olteanu, with the title Ring isomorphism of cyclic cyclotomic algebras, 22.08.2008.
  • Research visit: Andrei Marcus, 14-27.09.2008, prof. dr. Changchang Xi, Beijing Normal University. 3 talks given at the research seminar of Representation Theory, with the titles: Equivalences of categories in modular representation theory, Morita equivalences induced by bimodules over Hopf-Galois extensions, Characters and equivalence classes of central simple group graded algebras.
  • Research visit: Gabriela Olteanu, 4-20.09.2008, prof. dr. Ángel del Río, Universidad de Murcia, Spain.
  • Research visit: Tiberiu Coconet, Andrei Marcus, 03-10.10.2008, prof. dr. Erzsébet Horváth, Budapest Technical University; Algebra seminar talk given by Andrei Marcus, with the title Hopf-Galois extensions and an exact sequence for H-Picard groups, at the Mathematical Institute, Budapest, 06.10.2008.
  • Organization of conference: Algebra Symposium, Babes-Bolyai University Cluj-Napoca, May 15-16, 2009. Talks given by T. Coconet with the title Remarks on induction of G-algebras and skew group algebras in 15.05.2009, and by C. Todea with the title New stable elements in Hochschild cohomology algebra in 16.05.2009.
  • Research visit: Gabriela Olteanu, prof. dr. Ángel del Río, Universidad de Murcia, Spain, February 2009.
  • Research visit: Gabriela Olteanu, Vrije Universiteit Brussel (VUB), Belgium, 1-14 February 2009 and 1-30 August 2009, prof. dr. Eric Jespers; Algebra seminar talk given by Gabriela Olteanu, with the title A computational approach for the Wedderburn decomposition of group algebras.
  • Research visit: Tiberiu Coconet, Cosmin Todea, 24.08-4.09.2009, prof. dr. Markus Linckelmann, University of Aberdeen.
    •
    Results
  • Camelia Dicu, On the multiplicity module of a pointed group, Mathematica (Cluj), 50 (73), No.1 (2008), 31-37. 
  • Camelia Dicu, Metode modul-teoretice in studiul G-algebrelor si al grupurilor punctate, EFES Publishing House, Cluj-Napoca, 2008, 92pp, ISBN 978-606-526-002-3.
  • Gabriela Olteanu, Wedderburn decomposition of group algebras and applications, EFES Publishing House, Cluj-Napoca, 2008, 165pp, ISBN 978-973-7677-98-3.
  • Gabriela Olteanu, Computation and applications of Schur indices, Proceedings of the International Conference on Modules and Representation Theory (Cluj-Napoca, Romania, July 7-12, 2008), Cluj University Press, Cluj-Napoca, 2009, 149-157.
  • Septimiu Crivei, Gabriela Olteanu, GAP algorithms for finite abelian groups and applications, Carpathian J. Math. 24 (2008), no. 3, 310-316.
  • Allen Herman, Gabriela Olteanu, Ángel del Río, The Schur group of an abelian number field, J. Pure Appl. Algebra 213 (2009), 22-33.
  • Allen Herman, Gabriela Olteanu, Ángel del Río, The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras, Israel J. Math. 176 (2010), 401-418.
  • Andrei Marcus, Indecomposable modules over group graded skew algebras, Mathematica (Cluj), 50 (73), No.2 (2008), 191-195.
  • Andrei Marcus, Derived invariance of Clifford classes, J. Group Theory, 12 (2009), 83-94.
  • Andrei Marcus, Derived equivalences and  the  abelian defect group conjecture, Proceedings of the  International Conference on Modules and Representation Theory, ''Babes-Bolyai" University Cluj, 2008; Cluj University Press, 111-131.
  • Stefaan Caenepeel, Andrei Marcus, Hopf-Galois extensions and an exact sequence for H-Picard groups, J. Algebra, 323 (2010), 622-657.
  • Simion Breaz, Septimiu Crivei, Andrei Marcus, Proceedings of the International Conference on Modules and Representation Theory, ''Babes-Bolyai" University Cluj, 2008; Cluj University Press, 2009. ISBN 978-973-610-897-6.
  • Constantin Cosmin Todea, Remarks on generalized Brauer pairs, Mathematica (Cluj), 52 (75) (2010), 75-84.
  • Tiberiu Coconet, Remarks on induction of G-algebras and skew group algebras, to appear in Mathematica (Cluj).
    •
    Reports
  • Sinteza 2007
  • Sinteza 2008
  • Sinteza 2009
  • Monitorizare 2009 - prezentare
    •
    Contact

    e-mail:
        marcus at math.ubbcluj.ro
    fax:
        +40 264 591906
    mail:
        Andrei Marcus
        Babes-Bolyai University
        Faculty of Mathematics and Computer Science
        Str. Mihail Kogalniceanu nr. 1
        RO-400084 Cluj-Napoca
        Romania