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RepnDecomp

Decompose representations of finite groups into irreducibles

1.2.1

2 March 2022

Kaashif Hymabaccus
Email: kaashif@kaashif.co.uk
Homepage: https://kaashif.co.uk

Contents

1.1 Getting started with RepnDecomp

  1.1-1 Installation
  1.1-2 Note on what is meant by a representation
  1.1-3 API Overview

2 Isomorphisms between representations

2.1 Finding explicit isomorphisms

2.1-1 LinearRepresentationIsomorphism
2.1-2 LinearRepresentationIsomorphismSlow

2.2 Testing isomorphisms

2.2-1 AreRepsIsomorphic
2.2-2 IsLinearRepresentationIsomorphism

3 Algorithms for unitary representations

3.1 Unitarising representations

  3.1-1 UnitaryRepresentation
  3.1-2 IsUnitaryRepresentation
  3.1-3 LDLDecomposition

3.2 Decomposing unitary representations

3.2-1 IrreducibleDecompositionDixon

5 Computing decompositions of representations

5.1 Block diagonalizing

5.1-1 BlockDiagonalBasisOfRepresentation
5.1-2 BlockDiagonalRepresentation

5.2 Algorithms due to the authors

5.2-1 REPN_ComputeUsingMyMethod
5.2-2 REPN_ComputeUsingMyMethodCanonical

5.3 Algorithms due to Serre

  5.3-1 CanonicalDecomposition
  5.3-2 IrreducibleDecomposition
  5.3-3 IrreducibleDecompositionCollected
  5.3-4 REPN_ComputeUsingSerre

6 Centralizer (commutant) rings

6.1 Finding a basis for the centralizer

6.1-1 CentralizerBlocksOfRepresentation
6.1-2 CentralizerOfRepresentation

6.2 Using the centralizer for computations

6.2-1 ClassSumCentralizer
6.2-2 ClassSumCentralizerNC

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