Student Seminar in Combinatorics: Mathematical Software Group - PowerPoint PPT Presentation

Student Seminar in Combinatorics: Mathematical Software Group Theory and Symmetries Loreno Heer Seminar of Prof. Komei Fukuda December 02, 2014

Contents Mathematics December 02, 2014 Student Seminar in Combinatorics: Mathematical Software ETHZ Loreno Heer 3 References Further Examples Rubik’s Cube 2 Examples Data Structures 1 Overview of the GAP System Syntax Overview GAP Programming Language GAP Data Libraries Overview Examples of GAP Packages GAP Package System GAP Implementation Overview GAP Introduction 2 / 26

GAP Introduction GAP stands for G roups, A lgorithms and P rogramming Development of GAP began at Lehrstuhl D für Mathematik, RWTH-Aachen, under the leadership of Joachim Neubüser in 1985. GAP is free software licensed under the GNU General Public License Today development is international project. Developers: University of St. Andrews; RWTH Aachen; the Technische Universität Braunschweig; Colorado State University GAP is included in Sage GAP uses the GNU GMP library since version 4.5 Loreno Heer Student Seminar in Combinatorics: Mathematical Software ETHZ December 02, 2014 3 / 26

GAP Implementation Overview Contains for example the Small Groups Library : Groups of order up to December 02, 2014 Student Seminar in Combinatorics: Mathematical Software ETHZ Loreno Heer 4 Documentation the prime number decomposition of their orders. 2000 (except 1024) and some infinite series of groups characterised by 3 A library of group theoretical data : The GAP system consists of the following parts: language & user contributed packages extending this library 2 A library of GAP functions written entirely in the GAP programming A ”read-evaluate-loop” user interface Implementation of time-critical functions Automatic dynamic storage management An interpreter for the GAP programming language 1 A kernel written in C which provides 4 / 26

GAP Package System GAP packages Self-contained extensions to the core system which contain GAP code and their own documentation and may also contain data files or external programs to which the GAP code provides an interface. Packages can be submitted for a peer-review process. A package that has successfully passed this process is called an accepted package . All other packages are called deposited packages . Loreno Heer Student Seminar in Combinatorics: Mathematical Software ETHZ December 02, 2014 5 / 26

Examples of GAP Packages provide algorithms for fans and cones of polymake to other December 02, 2014 Student Seminar in Combinatorics: Mathematical Software ETHZ Loreno Heer theory and an interface to PARI/GP ( accepted package ). Alnuth by B. Assmann, A. Distler, B. Eick: Algebraic number packages ( deposited package ). PolymakeInterface by T. Baechler, S. Gutsche: A package to GRAPE by L. H. Soicher: Gr aph A lgorithms using Pe rmutation Singular ( deposited package ). Singular by M. Costantini, W. de Graaf: The GAP interface to automata ( accepted package ). Automata by M. Delgado, S. Linton, J. Morais: A package on ncurses interface ( deposited package ). Browse by T. Breuer, F. Lübeck: Browsing applications and groups ( accepted package ). 6 / 26

There are many more packages available for GAP... Loreno Heer Student Seminar in Combinatorics: Mathematical Software ETHZ December 02, 2014 7 / 26

ACE 5.1 (22/01/2012) by G. Gamble, A. Hulpke, G. Havas, C. Ramsay Advanced Coset Enumerator RDS 1.6 (16/02/2012) by M. Roeder A package for searching relative difference sets LiePRing 1.6 (08/11/2013) by M. Vaughan-Lee, B. Eick Database and algorithms for Lie p-rings MapClass 1.2 (24/04/2012) by A. James, K. Magaard, S. Shpectorov, H. Volklein A Package For Mapping Class Orbit Computation ModIsom 2.3.2 (04/07/2014) by B. Eick, A. Konovalov Computing automorphisms and checking isomorphisms for modular group algebras of finite p-groups Nilmat 1.2 (10/08/2007) by A. Detinko, B. Eick, D. Flannery Computing with nilpotent matrix groups nq 2.5.1 (02/04/2014) by M. Horn, W. Nickel Nilpotent Quotients of Finitely Presented Groups OpenMath 11.2.0 (16/11/2013) by M. Costantini, A. Konovalov, M. Nicosia, A. Solomon OpenMath functionality in GAP ParGAP 1.4.0 (17/11/2013) by G. Cooperman, P. Smith, A. Konovalov Parallel GAP permut 1.01 (15/04/2014) by A. Ballester-Bolinches, E. Cosme-Llópez, R. Esteban-Romero, R. Esteban-Romero A package to deal with permutability in finite groups Polenta 1.3.2 (01/04/2014) by B. Assmann, M. Horn Polycyclic presentations for matrix groups Polycyclic 2.11 (07/03/2013) by B. Eick, W. Nickel, M. Horn Computation with polycyclic groups QuaGroup 1.8 (16/08/2013) by W. A. de Graaf a package for doing computations with quantum groups RadiRoot 2.7 (09/04/2014) by A. Distler Roots of a Polynomial as Radicals RCWA 3.7.0 (21/07/2014) by S. Kohl Residue-Class-Wise Affine Groups Repsn 3.0.2 (25/08/2011) by V. Dabbaghian A GAP4 Package for constructing representations of finite groups liealgdb 2.1 (28/03/2010) by S. Cicalo’, W. de Graaf, C. Schneider A database of Lie algebras SCSCP 2.1.4 (17/11/2013) by A. Konovalov, S. Linton Symbolic Computation Software Composability Protocol in GAP simpcomp 2.0.0 (12/12/2013) by F. Effenberger, J. Spreer A GAP toolbox for simplicial complexes SONATA 2.6 (07/11/2012) by E. Aichinger, F. Binder, J. Ecker, P. Mayr, C. Nöbauer System of nearrings and their applications Sophus 1.23 (03/02/2006) by C. Schneider Computing in nilpotent Lie algebras SymbCompCC 1.2 (19/11/2011) by D. Feichtenschlager Computing with parametrised presentations for p-groups of fixed coclass toric 1.8 (03/05/2012) by D. Joyner toric varieties and some combinatorial geometry computations UnitLib 3.2.0 (11/11/2014) by A. Konovalov, E. Yakimenko Library of normalized unit groups of modular group algebras Wedderga 4.7.1 (07/11/2014) by O. Broche Cristo, A. Herman, A. Konovalov, G. Olteanu, A. Olivieri, A. del Rio, I. Van Gelder Wedderburn Decomposition of Group Algebras XGAP 4.23 (30/04/2012) by F. Celler, M. Neunhoeffer a graphical user interface for GAP And many more ... Loreno Heer Student Seminar in Combinatorics: Mathematical Software ETHZ December 02, 2014 LieRing 2.2 (11/12/2013) by S. Cicalo’, W. A. de Graaf finitely presented Lie rings LAGUNA 3.7.0 (11/11/2014) by V. Bovdi, A. Konovalov, R. Rossmanith, C. Schneider Lie AlGebras and UNits of group Algebras Alnuth 3.0.0 (26/10/2011) by B. Assmann, A. Distler, B. Eick Algebraic number theory and an interface to PARI/GP EDIM 1.3.2 (12/06/2013) by F. Lübeck Elementary Divisors of Integer Matrices ANUPQ 3.1.1 (19/11/2013) by G. Gamble, W. Nickel, E. O’Brien, M. Horn ANU p-Quotient Automata 1.13 (19/11/2011) by M. Delgado, S. Linton, J. Morais A package on automata AutPGrp 1.6 (29/05/2012) by B. Eick, E. O’Brien Computing the Automorphism Group of a p-Group Carat 2.1.4 (29/05/2012) by F. Gähler Interface to CARAT, a crystallographic groups package Circle 1.5.2 (01/02/2013) by A. Konovalov, P. Soules Adjoint groups of finite rings cohomolo 1.6 (23/06/2008) by D. Holt Cohomology groups of finite groups on finite modules Congruence 1.1.1 (28/10/2014) by A. Dooms, E. Jespers, A. Konovalov, H. Verrill Congruence subgroups of SL(2,Integers) CoReLG 1.02 (10/02/2014) by H. Dietrich, P. Faccin, W. A. de Graaf computation with real Lie groups Crime 1.4 (01/6/2011) by M. Bishop A GAP Package to Calculate Group Cohomology and Massey Products CRISP 1.3.8 (18/02/2014) by B. Höfling Computing with Radicals, Injectors, Schunck classes and Projectors Cryst 4.1.12 (10/10/2013) by B. Eick, F. Gähler, W. Nickel Computing with crystallographic groups Cubefree 1.13 (10/12/2012) by H. Dietrich Constructing the Groups of a Given Cubefree Order DESIGN 1.6 (23/11/2011) by L. H. Soicher The Design Package for GAP FactInt 1.5.3 (16/06/2011) by S. Kohl Advanced Methods for Factoring Integers kbmag 1.5 (06/01/2009) by D. Holt Knuth-Bendix on Monoids and Automatic Groups FGA 1.2.0 (20/11/2012) by C. Sievers Free Group Algorithms FORMAT 1.3 (05/26/2012) by B. Eick, C. R. Wright Computing with formations of finite solvable groups. Forms 1.2.2 (29/08/2011) by J. Bamberg, J. De Beule Sesquilinear and Quadratic FPLSA 1.1 (17/11/2003) by V. Gerdt, V. Kornyak Finitely Presented Lie Algebras GAPDoc 1.5.1 (23/02/2012) by F. Lübeck, M. Neunhöffer A Meta Package for GAP Documentation GBNP 1.0.1 (12/05/2010) by A. Cohen, J. Knopper computing Gröbner bases of noncommutative polynomials GRAPE 4.6.1 (17/05/2012) by L. H. Soicher GRaph Algorithms using PErmutation groups GrpConst 2.3 (29/05/2012) by H. U. Besche, B. Eick Constructing the Groups of a Given Order GUAVA 3.12 (21/05/2012) by R. Baart, T. Boothby, J. Cramwinckel, J. Fields, D. Joyner, R. Miller, E. Minkes, E. Roijackers, L. Ruscio, C. Tjhai a GAP package for computing with error-correcting codes HAP 1.10.15 (07/12/2013) by G. Ellis Homological Algebra Programming hecke 1.4 (02/07/2013) by D. Traytel Hecke - Specht 2.4 ported to GAP 4 IRREDSOL 1.2.4 (18/02/2014) by B. Höfling A Library of irreducible solvable linear groups over finite fields and of finite primivite soluble groups ITC 1.4 (06/01/2004) by V. Felsch, L. Hippe, J. Neubüser Interactive Todd-Coxeter 8 / 26