GAP (GroupRepresentationsForCAP) - Chapter 4: Representation Category of Groups

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

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

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

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

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.