Research project

    Categorical and combinatorial methods in representation theory
    Versiunea in limba romana
    Information
    Program: PN2 IDEI (Information on the program at UEFISCDI)
    Project type: PN-II-ID-PCE-2012-4 (Exploratory Research Projects)
    Contracting authority: UEFISCDI
    Contractor: ,,Babes-Bolyai" University Cluj-Napoca
    Project code: PN-II-ID-PCE-2012-4-0100
    Contract no: 51/02.09.2013
    Duration: 37 months, 01.09.2013-30.09.2016
    Research team
    Andrei Marcus, project director
    Simion Breaz, senior researcher
    Septimiu Crivei, senior researcher
    George Ciprian Modoi, senior researcher
    Gabriela Olteanu, senior researcher
    Csaba Szántó, senior researcher
    Tiberiu Coconet, postdoctoral researcher
    Emil Horobet, research assistant
    Abstract
    In this this project we focus on group algebras of finite groups, path algebras of quivers, and their representations. We also investigate generalizations of these algebras, and actions of groups on them, leading to gradings and Galois coverings. The research in this field is very intense on an international level, being motivated by many important open problems, and it has multiple ramifications and applications. We combine the conceptualization provided by category theory with combinatorics. Our main directions of investigation are as follows: Clifford theory of blocks, Morita and derived equivalences graded by groups with order divisible by p; a categorical view on Brauer-Clifford groups; character correspondences vs. derived equivalences in connection with conjectures of Alperin, McKay, Dade, Isaacs, Navarro, Uno, Turull; reduction to simple or quasisimple groups of the calculation of numerical invariants associated to blocks; derived equivalences and tilting, and their graded versions; stable derived categories, n-angulated categories, exact categories, adjoint functors and separability; gradings and Galois coverings; combinatorial description of the basic algebra of a skew group algebra.
    Activities
  • Research seminar: weekly, Thursday 11:00-15:00, UBB Mathematicum building, Algebra room.
  • Research visit: Szanto Csaba at Faculty of Mathematics, University din Stuttgart, Germania, 12-27.10.2013.
  • Research visit: Andrei Marcus at Alfred Rényi Mathematical Institute of the Hungarian Academy, Budapest, 15-19.11.2013.
  • Research visit: Szanto Csaba at Alfred Rényi Mathematical Institute of the Hungarian Academy, Budapest, 23-27.11.2013.
  • Research visit: Septimiu Crivei at Carlos III University, Madrid, Spain, 23-30.11.2013.
  • Research visit: Gabriela Olteanu at Carlos III University, Madrid, Spain, 23-30.11.2013.
  • Algebra seminar talk with the title Separable extensions of OP-algebras and Morita equivalences, given by Andrei Marcus at Alfred Rényi Mathematical Institute, Budapest, 18.11.2013.
  • Algebra seminar talk Equivalences of categories in modular representation theory, given by Andrei Marcus at Eotvos Lorand University, Budapest, 18.11.2013.
  • Algebra seminar talk New results on the Gabriel-Roiter measure, given by catre Csaba Szántó at Alfred Renyi Mathematical Institute, Budapest, 25.11.2013.
  • Research visit: Szanto Csaba at Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 4-12.07.2014.
  • Research visit: Tiberiu Coconet at Faculty of Mathematics, University of Jena, Germany, 22-29.09.2014.
  • Research visit: Szanto Csaba at Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 21-28.09.2014.
  • Research visit: Andrei Marcus at Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 06-11.11.2014.
  • Research visit: Szanto Csaba at Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 29.10-05.11.2014.
  • Research visit: Ciprian Modoi at Departamentul de Algebra, University Charles, Prague, 09-12.11.2014.
  • Research visit: Simion Breaz at Departamentul de Algebra, University Charles, Prague, 09-12.11.2014.
  • Invited talk given by Septimiu Crivei, First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, Bilbao, Spain, 30 iunie-4 iulie 2014, at the Rings, Modules, Categories and Applications session, with the title Maximal exact structures on additive categories.
  • Algebra seminar talk with the title: Ideal cotorsion pairs in triangulated categories I given by Ciprian Modoi 10.11.2014, Departament of Algebra, Charles University, Prague.
  • Algebra seminar talk with the title: Ideal cotorsion pairs in triangulated categories II given by Simion Breaz, 10.11.2014, Departament of Algebra, Charles University, Prague .
  • Talk given by Gabriela Olteanu, First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, Bilbao, June 30 - July 4, 2014.
  • Talk given by Gabriela Olteanu, Annual Scientific Conference of the Romanian Academic Economists from Abroad, Cluj-Napoca, August 18-22, 2014.
  • Algebra seminar talk with the title Hall Polynomials in euclidean casesgiven by Csaba Szanto, November 3, 2014, Alfred Renyi Mathematical Institute, Budapest, Seminar of Algebra.
  • Algebra seminar talk with the title Morita equivalences related to the Glauberman correspondence given by Andrei Marcus, November 10, 2014, Alfred Renyi Mathematical Institute, Budapest, Seminar of Algebra.
  • Algebra seminar talk, Andrei Marcus, 22.04.2015, University of Pécs, Hungary, Department of Mathematics. Talk title: Frobenius extensions, Hopf subalgebras, and induction.
  • Algebra seminar talk, Andrei Marcus, 23.04.2015, Hungarian Academy of Science, Pécs Branch. Talk title: Equivalences of categories in modular representation theory.
  • Talk given by Tiberiu Coconet, 29.06.2015, The Eighth Congress of Romanian Mathematicians, 26 Iunie 2015-1 Iulie 2015. Talk title: Module covers and the Green correspondence.
  • Talk given by Simion Breaz, The Eighth Congress of Romanian Mathematicians, Iasi, 26 iunie-1 iulie 2015. Talk title: Pure semisimple rings and direct products.
  • Research visit: Simion Breaz at Workshop: Silting theory and related topics, University of Verona, 2-4.07.2015.
  • Research visit: Andrei Marcus at Faculty of Mathematics, University of Jena, Germany, 21-26.07.2015.
  • Talk given by Simion Breaz, Some Trends in Algebra, Charles Univ. Prague, 1-4 septembrie 2015. Talk title: A Baer-Kaplansky Theorem for PID's.
  • Talk given by Ciprian Modoi at Program: Interactions between Representation Theory, Algebraic Topology and Commutative Algebra, Centre de Recerca Matemática, Universitat Autonoma de Barcelona: Workshop Bellaterra, Spain: Brave new Algebra: opening perspective, 20-24.04.2015. Talk title: Examples and counterexamples concerning Brown representability and its dual.
  • Talk given by Ciprian Modoi at 4-th International Eurasian Conference on Mathematical Sciences and Applications, Atena, Grecia 31.08-3.09 2015. Talk title: The solution of a matrix completion problem and nil-clean companion matrices.
  • Research visit: Szanto Csaba at Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 21.10-25.10.2015.
  • Simion Breaz, 11th Joint Conference on Mathematics and Computer Science, 20-22 mai 2016, Eger, Ungaria. Talk title: Nil-clean matrices.
  • Research visit: Szanto Csaba la Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 27.04-01.05.2016.
  • Simion Breaz, Conference on Rings and Polynomials, 3-8 iulie 2016, Technische Univ. Graz, Austria. Talk title: Pure semisimple rings and direct products.
  • Ciprian Modoi, Conference on Rings and Polynomials, 3-8 iulie 2016, Technische Universität Graz, Austria. Talk title: Nil-clean companion matrices.
  • Research visit: Szanto Csaba la Mathematical Institute Alfred Rényi of the Hungarian Academy, Budapest, 10.05-15.05.2016.
  • Ciprian Modoi, International Conference on Representation of Algebras (ICRA 2016), 10-19 august 2016, Syracuse University NY, SUA. Talk title: Derived equivalences induced by silting complexes.
  • Gabriela Olteanu, 11th Joint Conference on Mathematics and Computer Science, 20-22 mai 2016, Eger, Ungaria. Talk title: Primitive idempotents and group codes.
  • Andrei Marcus, Conference on Brauer groups, Hopf algebras and monoidal categories, 24-27 mai 2016, Torino, Italia. Talk title: Frobenius induction for algebras.
  • Andrei Marcus, Universitatea din Verona, Italia, 30 mai - 3 iunie 2016. Comunicare cu titlul: Morita equivalences in modular representation theory.
  • Tiberiu Coconet, Workshop Morita equivalence problems for blocks of finite groups, Ecole Polytechnique Fédérale de Lausanne (EPFL), 5-9 septembrie 2016.
  • Andrei Marcus, Workshop Endo-p-permutation and trivial source modules; Morita equivalence problems for blocks of finite groups, Ecole Polytechnique Fédérale de Lausanne (EPFL), 29 august - 9 septembrie 2016. Talk title: Group graded basic Morita equivalences.
    •
    Results
  • Tiberiu Coconet: Covering points in permutation algebras, Arch. Math. 100 (2013), 107-113.
  • Tiberiu Coconet and Constantin-Cosmin Todea: The extended Brauer quotient of N-interior G-algebras, Journal of Algebra 396 (2013) 10-17.
  • Dana Debora Glitia and Andrei Marcus: Field extensions and Clifford theory, Mathematica, Tome 55(78), No. 2, 2013 , 186-194.
  • Emil Horobet: Galois covering and smash product of skew categories, Acta Univ. Sapientiae, Mathematica, vol. 5, No. 2 (2013), 47-53.
  • Simion Breaz: Finite torsion-free rank endomorphism rings, Carpathian J. Math. 31 (2015), No. 1, 39-43.
  • Tiberiu Coconet: G-Algebras and Clifford Extensions of Points, Algebra Colloquium 21:4, (2014), 711-720.
  • Tiberiu Coconet: Restricting the Clifford extensions of a pointed group, Studia Univ. Babes-Bolyai Math. 59 (2014), No. 3, 269-277.
  • Tiberiu Coconet, Andrei Marcus: Group graded Hecke interior algebras. Mathematica 56 (79), No. 2, (2014), 117-126.
  • Septimiu Crivei: The Square-Cross Lemma, Bull. Math. Soc. Sci. Math. Roumanie 57 (105) (2014), 261-269.
  • G. C. Modoi, Constructing cogenerators in triangulated categories and Brown representability, J. Pure Appl. Algebra, 219 (2015), 3214-3224.
  • Gabriela Olteanu, I. Van Gelder: Construction of minimal non-abelian left group codes, Designs, Codes Cryptogr., 75 (2015), 359-373.
  • Gabriela Olteanu: Baer-Galois connections and applications, Carpathian J. Math., 30 (2014), No. 2, 225-229.
  • Csaba Szanto, Istvan Szollosi: Hall polynomials and the Gabriel-Roiter submodules of simple homogeneous modules, Bulletin of London Mathematical Society, 47 (2015), 206-216.
  • Simion Breaz, Jan Zemlicka: The defect functor of a homomorphism and direct unions, Algebras and Representation Theory 19 (2016), Issue 1, 181-208, DOI: 10.1007/s10468-015-9569-0, arXiv:1408.2016v2.
  • Simion Breaz, P. Danchev, Y. Zhou: Rings in which every element is either a sum or a diference of a nilpotent and an idempotent, J. Algebra Appl. (15, 1650148 (2016) online), DOI: 10.1142/S0219498816501486.
  • Tiberiu Coconet, Andrei Marcus, Module covers and the Green correspondence, Journal of Algebra 432 (2015), 62-71.
  • Dana-Debora Glitia, Andrei Marcus: Morita equivalences related to the Glauberman correspondence, Journal of Algebra 436 (2015), 17-34.
  • Csaba Szanto: Combinatorial aspects of extensions of Kronecker modules, Journal of Pure and Applied Algebra, 219 (2015), 4378-4391.
  • Septimiu Crivei, Arda Kör, Rickart and dual Rickart objects in abelian categories, Applied Categorical Structures, (online), DOI: 10.1007/s10485-015-9405-z.
  • Simion Breaz, George C. Modoi, Nil-clean companion matrices, Linear Algebra and its Applications 489 (2016), 50-60.
  • Gabriela Olteanu, I. Van Gelder: On Idempotents and the number of simple components of semisimple group algebras, Algebr. Represent. Theory 19 (2016), 315-333, DOI 10.1007/s10468-015-9575-2.
  • Simion Breaz, Grigore Calugareanu: Sums of nilpotent matrices, Linear and Multilinear Algebra (online 2016, DOI: 10.1080/03081087.2016.1167816).
  • Simion Breaz, Grigore Calugareanu: Strongly Inert Abelian Groups, Rendiconti del Seminario Matematico della Universita di Padova (online 2016).
  • Emil Horobet, An algorithm of constructing the basic algebra of a skew group algebra, MATH. REPORTS 18(68), no. 3 (2016), 403-416.
  • Csaba Szanto, Abel Lorinczi: The indecomposable preprojective and preinjective representations of the quiver Dn, Mathematica (Cluj), Tome 57 (80), No. 1-2 (2015).
  • M.K. Berktas, Septimiu Crivei, On complete pure-injectivity in locally finitely presented categories, Bull. Math. Soc. Sci. Math. Roumanie, to appear.
  • Septimiu Crivei, Gabriela Olteanu: Rickart and dual Rickart objects in abelian categories: transfer via functors, submitted.
  • Tiberiu Coconet, Andrei Marcus, Constantin-Cosmin Todea: Frobenius Induction for Algebras, Communications in Algebra, to appear.
    • Reports
    • Scientific Report 2013
    • Scientific Report 2014
    • Scientific Report 2013-2015
    • Scientific Report 2013-2016
    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