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# MajoranaAlgebras

## A package for constructing Majorana algebras and representations

1.5.1

9 December 2022

Markus Pfeiffer
Email: markus.pfeiffer@st-andrews.ac.uk
Homepage: https://markusp.morphism.de/

Madeleine Whybrow
Email: mlw10@ic.ac.uk
Homepage: https://madeleinewhybrow.wordpress.com
Address:
Department of Mathematics, Imperial College, South Kensington, SW7 2AZ

### Copyright

© 2018 by Markus Pfeiffer and Madeleine Whybrow

MajoranaAlgebras package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

### Contents

1 Introduction
1.1 A quick guide
1.2 Understanding the output
1.3 Info levels
1.3-1 InfoMajorana
2 Shapes of a Majorana representation
2.1 The shapes functions
2.1-1 ShapesOfMajoranaRepresentation
2.1-2 ShapesOfMajoranaRepresentationAxiomM8
2.1-3 MAJORANA_IsSixTranspositionGroup
2.1-4 MAJORANA_RemoveDuplicateShapes
3 Majorana representations
3.1 The main function
3.1-1 MajoranaRepresentation
3.2 The n-closed function
3.2-1 NClosedMajoranaRepresentation
4 Functions for calculating with Majorana representations
4.1 Calculating products
4.1-1 MAJORANA_AlgebraProduct
4.1-2 MAJORANA_InnerProduct
4.2 Basic functions
4.2-1 MAJORANA_IsComplete
4.2-2 MAJORANA_Dimension
4.2-3 MAJORANA_Eigenvectors
4.2-4 MAJORANA_Basis
4.2-5 MAJORANA_AdjointAction
4.3 The subalgebra structure
4.3-1 MAJORANA_Subalgebra
4.3-2 MAJORANA_IsJordanAlgebra
5 Functions for testing Majorana representations
5.1 The main function
5.1-1 MajoranaAlgebraTest
5.2 Other functions
5.2-1 MAJORANA_TestFrobeniusForm
5.2-2 MAJORANA_TestInnerProduct
5.2-3 MAJORANA_TestAxiomM2
5.2-4 MAJORANA_TestPrimitivity
6 Orbital Structures
6.1 Examples
6.1-1 IsOrbitalStructure
6.1-2 OrbitalStructure
6.1-3 OS_OrbitRepresentative
6.1-4 OS_CanonisingElement
6.1-5 OS_CanonisingElementAndRepresentative
6.1-6 OS_StabilizerOf
6.1-7 OrbitalRepresentative
6.1-8 AllOrbitalRepresentatives
6.1-9 OrbitalCanonizingElement
6.1-10 OrbitalCanonizingElementInverse
6.1-11 OrbitalTransversalIterator
6.1-12 UnorderedOrbitalRepresentative
6.1-13 AllUnorderedOrbitalRepresentatives
6.1-14 UnorderedOrbitalTransversalIterator
6.1-15 UnorderedOrbitalCanonizingElement
6.1-16 UnorderedOrbitalCanonizingElementInverse
7 Signed Permutations
7.1 Different Representations
7.2 Low-Level Descriptions
7.2-1 IsSignedPerm
7.2-2 ListSignedPerm
7.2-3 ListSignedPerm
7.2-4 SignedPerm
7.2-5 NewSignedPerm
7.2-6 NewSignedPerm
7.2-7 IsSignedPermRep
7.2-8 IsSignedPermListRep
7.2-9 OnPosPoints
7.2-10 LargestMovedPoint
7.2-11 RandomSignedPermList
7.2-12 RandomSignedPerm
Index
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