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Blocks of Character Tables

(Version 0.9.5)

Thomas Breuer
Email: sam@math.rwth-aachen.de
Homepage: http://www.math.rwth-aachen.de/~Thomas.Breuer

Copyright

© 2013–2022

This package may be distributed under the terms and conditions of the GNU Public License Version 3 or later, see http://www.gnu.org/licenses.

Contents

1 Introduction to the CTBlocks Package

1.1 Acknowledgements

2 Tutorial for the CTBlocks Package

2.1 Block invariants of covers of sporadic simple groups

2.2 More about abelian defect groups

2.3 Examples of block induction

2.3-1 Examples of block induction: Separating examples
2.3-2 Examples of block induction: The Mathieu groups \(M_{11}\) and \(M_{12}\), for \(p = 2\)

2.4 Examples about reality questions

2.4-1 Example: Strongly real classes of sporadic simple groups
2.4-2 Example: Strongly real blocks of sporadic simple groups

2.5 Examples about the Loewy length of centres of blocks

2.5-1 Example: Loewy lengths of centres of blocks of sporadic simple groups

3 Character theoretic functions for \(p\)-blocks

3.1 The character theoretic setup

3.2 Theoretical background of blocks of character tables

3.2-1 A number theoretic lemma

3.3 Block objects for character tables

  3.3-1 Block
  3.3-2 Defining attributes of block objects
  3.3-3 PrincipalBlock
  3.3-4 PBlocks
  3.3-5 Irr
  3.3-6 IBr
  3.3-7 DecompositionMatrix
  3.3-8 DimensionsOfDecompositionMatrix
  3.3-9 Dimension
  3.3-10 CentralCharacter
  3.3-11 ClassPositionsOfKernel
  3.3-12 IsFaithful
  3.3-13 CoefficientsOfOsimaIdempotent
  3.3-14 Technicalities of block objects for character tables

3.4 Defect, defect classes, and defect groups

  3.4-1 Defect
  3.4-2 DefectGroup
  3.4-3 NormalizerOfDefectGroup
  3.4-4 ClassPositionsOfDefectGroupOfClass
  3.4-5 ClassPositionsOfDefectGroupOfBlock
  3.4-6 IsClassWithNormalDefectGroup
  3.4-7 IsBlockWithNormalDefectGroup
  3.4-8 IsClassWithCyclicDefectGroup
  3.4-9 IsBlockWithCyclicDefectGroup
  3.4-10 TestBlockWithAbelianDefectGroup
  3.4-11 ClassPositionsOfDefectClasses
  3.4-12 ClassPositionsOfDefectClassesUpToGaloisConjugacy

3.5 Radical \(p\)-subgroups

  3.5-1 IsRadicalPSubgroup
  3.5-2 NormalizerOfRadicalPSubgroup
  3.5-3 RepresentativesRadicalPSubgroupsAndNormalizers

3.6 Chains of radical \(p\)-subgroups

3.6-1 RepresentativesChainsOfRadicalPSubgroupsAndNormalizers

3.7 \(p\)-weights

3.7-1 DescriptionOfPWeights

3.8 Block induction

  3.8-1 BrauerCorrespondent
  3.8-2 PRegularCorrespondent
  3.8-3 AlperinBurryCorrespondent
  3.8-4 WheelerCorrespondent

3.9 Reality questions about blocks

  3.9-1 IsRealClass
  3.9-2 IsStronglyRealClass
  3.9-3 IsRealBlock
  3.9-4 IsStronglyRealBlock

3.10 Selecting blocks according to their invariants

  3.10-1 AllPBlocks
  3.10-2 OnePBlock
  3.10-3 BlockInvariants
  3.10-4 DisplayBlockInvariants
  3.10-5 BrowseBlockInvariants
  3.10-6 Customizing the columns of block invariants overviews

3.11 Centres of \(p\)-blocks as algebras

  3.11-1 SCAlgebraCentreOfGroupAlgebra
  3.11-2 BlockDecompositionSCAlgebraCentreOfGroupAlgebra
  3.11-3 SCAlgebraCentreOfBlock

4.1 Generalized Straight Line Programs

  4.1-1 IsGeneralizedStraightLineProgram
  4.1-2 GeneralizedStraightLineProgram
  4.1-3 DataOfGeneralizedStraightLineProgram
  4.1-4 NrInputsOfGeneralizedStraightLineProgram
  4.1-5 NrOutputsOfGeneralizedStraightLineProgram
  4.1-6 ResultOfGeneralizedStraightLineProgram
  4.1-7 EquivalentStraightLineProgram

4.2 Miscellaneous

  4.2-1 PRegularTable
  4.2-2 PrintOverviewOfDefectOneNormalizers
  4.2-3 FrobeniusCharacterValueExt
  4.2-4 ReductionToFiniteField
  4.2-5 InfoBlocks

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