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### 1 Associators

#### 1.1 Introduction

Let $$G$$ be a finite group and let $$G$$-mod be a skeletal version of the monoidal category of finite dimensional complex representations of $$G$$. The purpose of these GAP methods is the computation of the associators of $$G$$-mod.

#### 1.2 Quickstart

The following commands compute the associator of $$D_8$$ and write all data necessary for the reproducibility of the computation to files with the prefix "D8".

gap> G := DihedralGroup( 8 );;
gap> ComputeAssociator( G, true, true, false );;
gap> path := Filename( DirectoryTemporary( ), "D8" );;
gap> WriteAssociatorComputationToFiles( path );


#### 1.3 Read, Write, and Display

The following intermediate steps of the associator computation can be read from/written to files.

• Irreducible representations of a finite group given by matrices (Data 1).

• Decomposition isomorphisms of tensor products into direct sums of irreducibles (Data 2).

Furthermore, the following data can be written to files.

• A database key for the AssociatorsDatabase/DatabaseKeys.g file.

• The final result, namely the associator (Data 3).

Data 1 and Data 2 involve choices and thus are subject to changes in further versions of this package. However, the process Data 2 -> Data 3 is a mathematical function and thus stable. For reproducibility, it is recommended to store all three data. To facilitate this task, use the function WriteAssociatorComputationToFiles.

##### 1.3-1 WriteDatabaseKeysToFile
 ‣ WriteDatabaseKeysToFile( s ) ( operation )

Returns: nothing

The argument is a filename $$s$$. This operation writes the database keys computed by the last call of InitializeGroupData to the corresponding file.

##### 1.3-2 WriteRepresentationsDataToFile
 ‣ WriteRepresentationsDataToFile( s ) ( operation )

Returns: nothing

The argument is a filename $$s$$. This operation writes the representations computed by the last call of InitializeGroupData to the corresponding file.

##### 1.3-3 WriteSkeletalFunctorDataToFile
 ‣ WriteSkeletalFunctorDataToFile( s ) ( operation )

Returns: nothing

The argument is a filename $$s$$. This operation writes the skeletal functor data computed by the last call of SkeletalFunctorTensorData to the corresponding file.

##### 1.3-4 WriteAssociatorDataToFile
 ‣ WriteAssociatorDataToFile( s ) ( operation )

Returns: nothing

The argument is a filename $$s$$. This operation writes the associator data of the initialized group to the corresponding file. You have to call AssociatorForSufficientlyManyTriples first.

##### 1.3-5 WriteAssociatorComputationToFiles
 ‣ WriteAssociatorComputationToFiles( s ) ( operation )

Returns: nothing

Only call this function if you did a whole associator computation first (e.g. using ComputeAssociator). The argument is a string $$s$$. This function writes 4 files:

• $$s$$Key.g: A file for the database key of the associator computation.

• $$s$$Reps.g: A file containing the irreducible representations used for the associator computation.

• $$s$$Dec.g: A file for the tensor decompositions used for the associator computation.

• $$s$$Ass.g or $$s$$AssD.g: A file containing the computed associator. The suffix $$D$$ is used if the associator was not computed for all triples.

 ‣ ReadDatabaseKeys( s ) ( operation )

Returns: a list

The argument is a filename $$s$$ of a file written by WriteDatabaseKeysToFile. The output is a list [ group, conductor, position of trivial character, field, category ].

 ‣ ReadRepresentationsData( s_1, s_2 ) ( operation )

Returns: a list

The arguments are a filename $$s_1$$ of a file written by WriteDatabaseKeysToFile, and a filename $$s_2$$ of a file written by WriteRepresentationsDataToFile. The output is a list [ ,number of irreducibles, irreducibles, representations given by images of generators, inverses of these images, vector space objects for the irreducibles ].

 ‣ ReadSkeletalFunctorData( s_1, s_2 ) ( operation )

Returns: a list

The arguments are a filename $$s_1$$ of a file written by WriteDatabaseKeysToFile, and a filename $$s_2$$ of a file written by WriteSkeletalFunctorDataToFile. The output is a list [ irreducibles, skeletal functor tensor data, vector space objects for the irreducibles ].

#### 1.4 Computing associators

##### 1.4-1 InitializeGroupData
 ‣ InitializeGroupData( G ) ( operation )

Returns: a list

The argument is a group $$G$$. This method calls InitializeGroupData( G, false ).

##### 1.4-2 InitializeGroupData
 ‣ InitializeGroupData( G, b ) ( operation )

Returns: a list

The arguments are a group $$G$$ and a boolean $$b$$. The output is a list [ generators of $$G$$ ,number of irreducibles, irreducibles, representations given by images of generators, inverses of these images, vector space objects for the irreducibles ]. Furthermore, this method stores the database key, which can be written using WriteDatabaseKeysToFile. If $$b$$ is true, then the id of the group in the database key is given by its string, otherwise it is given by its id in the SmallGroupLibrary.

 ‣ InitializeGroupDataDixon( G ) ( operation )

Returns: a list

The argument is a group $$G$$. This method calls InitializeGroupDataDixon( G, false ).

 ‣ InitializeGroupDataDixon( G, b ) ( operation )

Returns: a list

The arguments are a group $$G$$ and a boolean $$b$$. This method does the same as InitializeGroupData, but uses IrreducibleRepresentationsDixon for affording irreducible representations.

##### 1.4-5 InitializeGroupData
 ‣ InitializeGroupData( arg1, arg2, arg3 ) ( operation )

##### 1.4-6 SkeletalFunctorTensorData
 ‣ SkeletalFunctorTensorData( ) ( operation )

Returns: a list

There is no argument. This methods calls SkeletalFunctorTensorData with the output of the last call of InitializeGroupData or InitializeGroupDataDixon.

##### 1.4-7 SkeletalFunctorTensorData
 ‣ SkeletalFunctorTensorData( l ) ( operation )

Returns: a list

The argument is a list $$l$$ which is the output of InitializeGroupData, InitializeGroupDataDixon, or ReadRepresentationsData. The output is a triple $$[t_1,t_2,t_3]$$. $$t_1$$ is the list of all characters of $$G$$. $$t_2$$ is a list such that the $$(i,j)$$-th entry, where $$i,j$$ range from 1 to the number of irreducibles, is a pair of mutual inverse morphisms $$[\alpha, \alpha^{-1}]$$, and $$\alpha$$ is a decomposition isomorphism $$\bigoplus_{\chi \in \mathrm{Irr}(G)}V_{\chi}^{n_{\chi}} \rightarrow V_i \otimes V_j$$. $$t_3$$ is a list of vector space objects for the irreducibles.

##### 1.4-8 AssociatorDataFromSkeletalFunctorTensorData
 ‣ AssociatorDataFromSkeletalFunctorTensorData( a, b, c, l ) ( operation )

Returns: a list

The arguments are integers $$a,b,c$$ and a list $$l$$ which is the output of SkeletalFunctorTensorData. The output is a list containing homalg matrices representing the components of the associator of $$V_a, V_b, V_c$$, where the numbers correspond to the enlisting of the irreducible characters given by $$l$$.

##### 1.4-9 AssociatorForSufficientlyManyTriples
 ‣ AssociatorForSufficientlyManyTriples( ) ( operation )

Returns: a list

There is no argument. This methods calls AssociatorForSufficientlyManyTriples with the output of the last call of SkeletalFunctorTensorData and false.

##### 1.4-10 AssociatorForSufficientlyManyTriples
 ‣ AssociatorForSufficientlyManyTriples( l, b ) ( operation )

Returns: a list

The arguments are a list $$l$$ which is the output of SkeletalFunctorTensorData, and a boolean $$b$$. The output is a list of lists $$L$$ such that $$L[a][b][c]$$ contains the associator computed by AssociatorDataFromSkeletalFunctorTensorData(a,b,c). If $$b$$ is true, then $$a,b,c$$ ranges through all possible triples, otherwise, $$a,b,c$$ are computed for so many triples such that the others can be obtained using braidings.

##### 1.4-11 ComputeAssociator
 ‣ ComputeAssociator( G, b_1 ) ( operation )

Returns: a list

The arguments are a group $$G$$, and a boolean $$b_1$$. The output is ComputeAssociator( G, b_1, false, true ).

##### 1.4-12 ComputeAssociator
 ‣ ComputeAssociator( G, b_1, b_2 ) ( operation )

Returns: a list

The arguments are a group $$G$$, and two booleans $$b_1$$, $$b_2$$. The output is ComputeAssociator( G, b_1, b_2, true ).

##### 1.4-13 ComputeAssociator
 ‣ ComputeAssociator( G, b_1, b_2, b_3 ) ( operation )

Returns: a list

The arguments are a group $$G$$, and three booleans $$b_1$$, $$b_2$$, $$b_3$$. The output is a list $$l$$ whose $$(a,b,c)$$-th entry contains a string representing the associator of the objects $$V_a, V_b, V_c$$ in a skeleton of the representation category of $$G$$, where $$V_{\ast}$$ are irreducible representations corresponding to the ordering of the irreducible characters Irr($$G$$). If $$b_1$$ is true, this method uses IrreducibleRepresentationsDixon, otherwise it uses IrreducibleAffordingRepresentation. If $$b_2$$ is true, the associators are computed for all possible triples $$a,b,c$$, otherwise only for sufficiently many such that the others can be reproduced using the braiding in the representation category. If $$b_3$$ is true, then the id of the group in the database key is given by its string, otherwise it is given by its id in the SmallGroupLibrary. This last boolean is relevant only if you want to write the computed associators to files (e.g. using WriteAssociatorComputationToFiles).

#### 1.5 Technical functions

##### 1.5-1 SetInfoLevelForAssociatorComputations
 ‣ SetInfoLevelForAssociatorComputations( l ) ( operation )

Returns: nothing

The argument is an integer $$l$$. If $$l > 0$$, then the functions for computing associators provide information during the computation. This is useful in cases where the computation may take a long time.

##### 1.5-2 DefinedOverCyclotomicField
 ‣ DefinedOverCyclotomicField( n, f ) ( operation )

Returns: a boolean

The arguments are an integer $$n$$ and a group homomorphism $$f$$ whose images are matrices. The output is true if the entries of the images of $$f$$ lie in a cyclotomic field generated by a primitive $$n$$-th root of unity, false otherwise.

##### 1.5-3 GroupReperesentationByImages
 ‣ GroupReperesentationByImages( G, L ) ( operation )

Returns: a group homomorphism

The arguments are a group $$G$$ with generators $$g_1, \dots, g_n$$ and a list $$L = [ l_1, \dots, l_n ]$$. The output is the group homomorphism from $$G$$ to the group generated by the elements of $$L$$, mapping $$g_i$$ to $$l_i$$.

##### 1.5-4 DiagonalizationTransformationOfBraiding
 ‣ DiagonalizationTransformationOfBraiding( e ) ( attribute )

Returns: an invertible endomorphism in $$\mathrm{Hom}(V,V)$$

The argument is an endomorphism $$e \in \mathrm{Hom}(V,V)$$ of vector spaces whose minimal polynomial divides $$x^2 - 1$$. The output is an invertible endomorphism $$t$$ such that $$t^{-1} \circ e \circ t$$ is a diagonal matrix.

##### 1.5-5 AffordAllIrreducibleRepresentations
 ‣ AffordAllIrreducibleRepresentations( G ) ( operation )

Returns: a list

The argument is a group $$G$$. The output is a list of all irreducible representations of $$G$$ using the command IrreducibleAffordingRepresentation.

##### 1.5-6 AffordAllIrreducibleRepresentationsDixon
 ‣ AffordAllIrreducibleRepresentationsDixon( G ) ( operation )

Returns: a list

The argument is a group $$G$$. The output is a list of all irreducible representations of $$G$$ using the command IrreducibleRepresentationsDixon.

##### 1.5-7 DefaultFieldForListOfRepresentations
 ‣ DefaultFieldForListOfRepresentations( L ) ( operation )

Returns: a GAP field

The argument is a list $$L$$ of representations of a group $$G$$. The output is a field over which all representations are defined simultaniously.

##### 1.5-8 RewriteMatrixInCyclotomicGenerator
 ‣ RewriteMatrixInCyclotomicGenerator( M, n ) ( operation )

Returns: a matrix

The arguments are a matrix $$M$$ and an integer $$n$$. The output is a matrix $$N$$ in $$Q[\epsilon]$$. Substituting an $$n$$-th root of unity for $$\epsilon$$ in $$N$$ yields $$M$$.

 ‣ InternalHomToTensorProductAdjunctMorphismTemp( b, c, g ) ( operation )

Returns: a morphism in $$\mathrm{Hom}(a \otimes b, c)$$.

The arguments are objects $$b,c$$ and a morphism $$g: a \rightarrow \mathrm{\underline{Hom}}(b,c)$$. The output is a morphism $$f: a \otimes b \rightarrow c$$ corresponding to $$g$$ under the tensor hom adjunction.

##### 1.5-10 HomalgMatrixAsString
 ‣ HomalgMatrixAsString( M ) ( operation )

Returns: a string

The argument is a homalg matrx $$M$$. The output is a string consisting of the elements of $$M$$, seperated by commas.

##### 1.5-11 DataFromSkeletalFunctorTensorDataAsStringList
 ‣ DataFromSkeletalFunctorTensorDataAsStringList( l ) ( operation )

Returns: a list of strings and empty entries

The argument is a list $$l$$ of homalg matrices. In $$l$$, empty entries are allowed. The output is a list where each non-empty entry of $$l$$ is converted to a string using HomalgMatrixAsString.

##### 1.5-12 AsVectorSpaceMorphism
 ‣ AsVectorSpaceMorphism( M ) ( attribute )

Returns: a vector space morphism

The argument is a homalg matrix $$M$$. The output is a vector space morphism whose underlying matrix is given by $$M$$.

##### 1.5-13 CreateEndomorphismFromString
 ‣ CreateEndomorphismFromString( V, s ) ( operation )

Returns: a vector space morphism

The arguments are a vector space object $$V$$ and a string $$s$$ consisting of $$\mathrm{dim}(V)^2$$ elements of the ground field of $$V$$. The output is a vector space endomorphism $$V \rightarrow V$$ defined by $$s$$.

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