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### 4 Representation Category of Groups

#### 4.1 Introduction

For a finite group $$G$$, the following methods provide computational tools for working with $$G$$-mod, a skeletal version of the monoidal category of finite dimensional complex representations of $$G$$, and with $$G-\mathbb{Z}$$-mod, a skeletal version of the monoidal category of finite dimensional complex representations of $$G$$ equipped with a degree in $$\mathbb{Z}$$.

#### 4.2 Quickstart

The following commands construct the category $$D_8$$-mod, the unique object $$v$$ corresponding to the irreducible character of degree 2, and perform some computations.

gap> RepG := RepresentationCategory( 8, 3 );
The representation category of Group( [ f1, f2, f3 ] )
gap> G := UnderlyingGroupForRepresentationCategory( RepG );
<pc group of size 8 with 3 generators>
gap> StructureDescription( G );
"D8"
gap> c := First( Irr( G ), i -> Degree( i ) = 2 );
Character( CharacterTable( D8 ), [ 2, 0, 0, -2, 0 ] )
gap> v := RepresentationCategoryObject( c, RepG );
1*(x_5)
gap> Dimension( v );
2
gap> Display( AssociatorLeftToRight( v, v, v ) );
Component: (x_5)

1/2,-1/2,1/2, 1/2,
1/2,-1/2,-1/2,-1/2,
1/2,1/2, 1/2, -1/2,
1/2,1/2, -1/2,1/2

A morphism in Category of matrices over Q
------------------------
gap> Display( Braiding( v, v ) );
Component: (x_1)

1

A morphism in Category of matrices over Q
------------------------
Component: (x_2)

1

A morphism in Category of matrices over Q
------------------------
Component: (x_3)

1

A morphism in Category of matrices over Q
------------------------
Component: (x_4)

-1

A morphism in Category of matrices over Q
------------------------
gap> alpha := IdentityMorphism( TensorProductOnObjects( v, v ) ) + Braiding( v, v );
<A morphism in The representation category of Group( [ f1, f2, f3 ] )>
gap> CokernelObject( alpha );
1*(x_4)
gap> TensorUnit( RepG );
1*(x_1)


#### 4.3 Constructors

##### 4.3-1 RepresentationCategory
 ‣ RepresentationCategory( G ) ( attribute )

Returns: a Cap category

The argument is a group $$G$$. The output is the Cap category $$G$$-mod. This method uses $$\texttt{String( G )}$$ as an identifier of $$G$$.

##### 4.3-2 RepresentationCategory
 ‣ RepresentationCategory( o, n ) ( operation )

Returns: a Cap category

The arguments are 2 integers $$o,n$$. The output is the Cap category $$G$$-mod, where $$G$$ is the group of order $$o$$ corresponding to the SmallGroupLibrary identification number $$n$$.

##### 4.3-3 RepresentationCategoryObject
 ‣ RepresentationCategoryObject( L, C ) ( operation )

Returns: an object in $$G$$-mod

There are 2 arguments. The first argument is a list $$L = [ [ n_1, c_1 ], \dots, [ n_l, c_l ] ]$$ consisting of non-negative integers $$n_i$$ and characters $$c_i$$ of the same group. Alternatively, the first argument can simply be an irreducible character c, which will be then interpreted as giving the input $$[ [ 1, c ] ]$$. The second argument is the Cap category $$C = G$$-mod. The output is the unique object in $$G$$-mod having $$L$$ as its character decomposition.

##### 4.3-4 RepresentationCategoryObject
 ‣ RepresentationCategoryObject( c, C, str ) ( operation )

Returns: an object in $$G$$-mod

There are 3 arguments. The first argument is an irreducible character c. The second argument is the CAP category $$C = G$$-mod. The third argument is a string used as follows: SetString( GIrreducibleObject( c ), str ). The output is the unique object in $$G$$-mod having $$[ [ 1, c ] ]$$ as its character decomposition.

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