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GroupRepresentationsForCAP

Skeletal category of group representations for CAP

2024.09-01

10 September 2024

Sebastian Posur
Email: sebastian.posur@uni-siegen.de
Homepage: https://sebastianpos.github.io
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

Contents

1.1 Introduction

1.3 Read, Write, and Display

  1.3-1 WriteDatabaseKeysToFile
  1.3-2 WriteRepresentationsDataToFile
  1.3-3 WriteSkeletalFunctorDataToFile
  1.3-4 WriteAssociatorDataToFile
  1.3-5 WriteAssociatorComputationToFiles
  1.3-6 ReadDatabaseKeys
  1.3-7 ReadRepresentationsData
  1.3-8 ReadSkeletalFunctorData

1.4 Computing associators

  1.4-1 InitializeGroupData
  1.4-2 InitializeGroupData
  1.4-3 InitializeGroupDataDixon
  1.4-4 InitializeGroupDataDixon
  1.4-5 InitializeGroupData
  1.4-6 SkeletalFunctorTensorData
  1.4-7 SkeletalFunctorTensorData
  1.4-8 AssociatorDataFromSkeletalFunctorTensorData
  1.4-9 AssociatorForSufficientlyManyTriples
  1.4-10 AssociatorForSufficientlyManyTriples
  1.4-11 ComputeAssociator
  1.4-12 ComputeAssociator
  1.4-13 ComputeAssociator

1.5 Technical functions

  1.5-1 SetInfoLevelForAssociatorComputations
  1.5-2 DefinedOverCyclotomicField
  1.5-3 GroupReperesentationByImages
  1.5-4 DiagonalizationTransformationOfBraiding
  1.5-5 AffordAllIrreducibleRepresentations
  1.5-6 AffordAllIrreducibleRepresentationsDixon
  1.5-7 DefaultFieldForListOfRepresentations
  1.5-8 RewriteMatrixInCyclotomicGenerator
  1.5-9 InternalHomToTensorProductAdjunctMorphismTemp
  1.5-10 HomalgMatrixAsString
  1.5-11 DataFromSkeletalFunctorTensorDataAsStringList
  1.5-12 CreateEndomorphismFromString

2 Semisimple Categories

2.1 Introduction

2.2 Constructors

  2.2-1 SemisimpleCategory
  2.2-2 SemisimpleCategory
  2.2-3 SemisimpleCategoryMorphism
  2.2-4 SemisimpleCategoryMorphismSparse
  2.2-5 ComponentInclusionMorphism
  2.2-6 ComponentProjectionMorphism
  2.2-7 SemisimpleCategoryObject
  2.2-8 SemisimpleCategoryObjectConstructorWithFlatList

  2.3-1 MembershipFunctionForSemisimpleCategory
  2.3-2 UnderlyingCategoryForSemisimpleCategory
  2.3-3 UnderlyingFieldForHomalgForSemisimpleCategory
  2.3-4 GivenObjectFilterForSemisimpleCategory
  2.3-5 GivenMorphismFilterForSemisimpleCategory
  2.3-6 SemisimpleCategoryMorphismList
  2.3-7 UnderlyingFieldForHomalg
  2.3-8 SemisimpleCategoryObjectList
  2.3-9 SemisimpleCategoryObjectListWithActualObjects
  2.3-10 Support
  2.3-11 UnderlyingFieldForHomalg
  2.3-12 Dimension

  2.4-1 Component
  2.4-2 NormalizeSemisimpleCategoryObjectList
  2.4-3 Multiplicity
  2.4-4 Component
  2.4-5 TestPentagonIdentity
  2.4-6 TestPentagonIdentityForAllQuadruplesInList
  2.4-7 TestBraidingCompatability
  2.4-8 TestBraidingCompatabilityForAllTriplesInList
  2.4-9 TestZigZagIdentitiesForDual
  2.4-10 TestZigZagIdentitiesForDualForAllObjectsInList

2.5 GAP Categories

2.5-1 IsSemisimpleCategoryMorphism
2.5-2 IsSemisimpleCategoryObject

3 Irreducible Objects

3.1 Introduction

3.2 Constructors

3.2-1 GIrreducibleObject
3.2-2 GZGradedIrreducibleObject

  3.3-1 UnderlyingCharacter
  3.3-2 UnderlyingGroup
  3.3-3 UnderlyingCharacterTable
  3.3-4 UnderlyingIrreducibleCharacters
  3.3-5 UnderlyingCharacterNumber
  3.3-6 Dimension
  3.3-7 Dual
  3.3-8 UnderlyingCharacter
  3.3-9 UnderlyingDegree
  3.3-10 UnderlyingGroup
  3.3-11 UnderlyingCharacterTable
  3.3-12 UnderlyingIrreducibleCharacters
  3.3-13 UnderlyingCharacterNumber
  3.3-14 Dimension
  3.3-15 Dual

3.4-1 IsYieldingIdentities
3.4-2 IsYieldingIdentities

  3.5-1 Multiplicity
  3.5-2 \*
  3.5-3 AssociatorFromData
  3.5-4 ExteriorPower
  3.5-5 Multiplicity
  3.5-6 \*
  3.5-7 AssociatorFromData
  3.5-8 ExteriorPower

4 Representation Category of Groups

4.1 Introduction

4.3 Constructors

  4.3-1 RepresentationCategory
  4.3-2 RepresentationCategory
  4.3-3 RepresentationCategoryObject
  4.3-4 RepresentationCategoryObject

5.1 Helper functions

5.1-1 LaTeXStringOfSemisimpleCategoryObjectList

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