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<Chapter Label=
"Chapter_Introduction">
<Heading>Introduction</Heading>
<Section Label=
"Chapter_Introduction_Section_Getting_started_with_RepnDecomp">
<Heading>Getting started with RepnDecomp</Heading>
This package allows computations of various decompositions of a
representation <Math>\rho : G \to GL(V)</Math> where <Math>G</Math> is finite and <Math>V</Math> is
a finite-dimensional <Math>\mathbb{C}</Math>-vector space.
<P/>
<Subsection Label=
"Chapter_Introduction_Section_Getting_started_with_RepnDecomp_Subsection_Installation">
<Heading>Installation</Heading>
<P/>
To install this package, refer to the installation instructions in
the README file in the source code. It is located here:
<
URL>
https://github.com/gap-packages/RepnDecomp/blob/master/README.md</
URL>.
</Subsection>
<Subsection Label=
"Chapter_Introduction_Section_Getting_started_with_RepnDecomp_Subsection_Note_on_what_is_meant_by_a_representation">
<Heading>Note on what is meant by a representation</Heading>
Throughout this documentation, mathematical terminology is used
e.g. representation. It is clear what is meant mathematically, but
it is not entirely clear what is meant in terms of GAP types - what
are you supposed to pass in when I say
"pass in a representation".
Occasionally I will not even mention what we are passing in and
assume the reader knows that <A>rho</A> or <Math>\rho</Math> refers to a
representation.
A representation we can use is, in GAP, a homomorphism from a
finite group to a matrix group where all matrices have coefficients
in a cyclotomic field (<Code>Cyclotomics</Code> is the union of all such
fields in GAP). You can check whether something you want to pass is
suitable with the function <Ref
Attr=
"IsFiniteGroupLinearRepresentation" Label=
"for IsGroupHomomorphism"/>.
<P/>
Here
's an example of a representation <A>rho</A> in GAP:
<P/>
<Example>
gap> G := SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> images := List(GeneratorsOfGroup(G), g -> PermutationMat(g, 3));
[ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] ]
gap> rho := GroupHomomorphismByImages(G, Group(images));
[ (1,2,3), (1,2) ] -> [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] ]
</Example>
<P/>
</Subsection>
<Subsection Label=
"Chapter_Introduction_Section_Getting_started_with_RepnDecomp_Subsection_API_Overview">
<Heading>API Overview</Heading>
<P/>
The algorithms implemented can be divided into two groups: methods
due to Serre from his book Linear Representations of Finite Groups,
and original methods due to the authors of this package.
<P/>
The default is to use the algorithms due to Serre. If you pass the
option <Code>method :=
"alternate"</Code> to a function, it will use the
alternate method. Passing the option <Code>parallel</Code> will try to compute
in parallel as much as possible. See the individual functions for
options you can pass.
<P/>
The main functions implemented in this package are:
<P/>
For decomposing representations into canonical and irreducible
direct summands:
<P/>
<List>
<Item>
<Ref Func=
"CanonicalDecomposition" />
</Item>
<Item>
<Ref Func=
"IrreducibleDecomposition" />
</Item>
<Item>
<Ref Func=
"IrreducibleDecompositionCollected" />
</Item>
</List>
<P/>
For block diagonalising representations:
<P/>
<List>
<Item>
<Ref Func=
"BlockDiagonalBasisOfRepresentation" />
</Item>
<Item>
<Ref Func=
"BlockDiagonalRepresentation" />
</Item>
</List>
<P/>
For computing centraliser rings:
<P/>
<List>
<Item>
<Ref Func=
"CentralizerBlocksOfRepresentation" />
</Item>
<Item>
<Ref Func=
"CentralizerOfRepresentation" />
</Item>
</List>
<P/>
For testing isomorphism and computing isomorphisms (intertwining
operators) between representations:
<P/>
<List>
<Item>
<Ref Func=
"LinearRepresentationIsomorphism" />
</Item>
<Item>
<Ref Func=
"AreRepsIsomorphic" />
</Item>
<Item>
<Ref Func=
"IsLinearRepresentationIsomorphism" />
</Item>
</List>
<P/>
For testing unitarity of representations and the unitarisation of
representations:
<P/>
<List>
<Item>
<Ref Func=
"UnitaryRepresentation" />
</Item>
<Item>
<Ref Func=
"IsUnitaryRepresentation" />
</Item>
</List>
</Subsection>
</Section>
</Chapter>
