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<!-- This is an automatically generated file. --> <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_Su <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_Su <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 rho 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_Su <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> ¤ Dauer der Verarbeitung: 0.0 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28