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.