GAP (MajoranaAlgebras) - Chapter 2: Shapes of a Majorana representation

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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

### 2 Shapes of a Majorana representation

#### 2.1 The shapes functions

##### 2.1-1 ShapesOfMajoranaRepresentation

`‣ ShapesOfMajoranaRepresentation`( G, T ) | ( function )

Returns: a record with a component shapes

Takes a group G and a G-invariant set of generating involutions T. Returns a list of possible shapes of a Majorana Representation of the form (G,T,V) that is stored in the shapes component of the output.

##### 2.1-2 ShapesOfMajoranaRepresentationAxiomM8

`‣ ShapesOfMajoranaRepresentationAxiomM8`( G, T ) | ( function )

Returns: a record with a component shapes

Performs exactly the same function as `ShapesOfMajoranaRepresentation` (2.1-1) but gives only those shapes at obey axiom M8. That is to say, we additionally assume that if \(t,s \in T\) such that \(|ts| = 2\) then the dihedral subalgebra \(\langle \langle a_t, a_s \rangle \rangle\) is of type \(2A\) if and only if \(ts \in T\) (and otherwise is of type \(2B\)).

##### 2.1-3 MAJORANA_IsSixTranspositionGroup

`‣ MAJORANA_IsSixTranspositionGroup`( G, T ) | ( function )

Returns: true if (G,T) is a 6-transposition group, otherwise returns false

For a group G and a subset T of G, returns true if all of the following conditions are satisfied: *T is a set of involutions that generate G; *T is closed under conjugation by G; *the order of the product of two elements of T is at most 6.

##### 2.1-4 MAJORANA_RemoveDuplicateShapes

`‣ MAJORANA_RemoveDuplicateShapes`( input ) | ( function )

If an automorphism of the group G stabilises the set T then it induces an action on the pairs of elements of T and therefore on the shapes of a possible Majorana representation of the form (G,T,V). If one shape is mapped to another in this way then their corresponding algebras must be isomorphic.

This function takes the record input as produced by the function `ShapesOfMajoranaRepresentation` (2.1-1) or `ShapesOfMajoranaRepresentationAxiomM8` (2.1-2) and replaces input.shapes with a list of shapes such that no two can be mapped to each other by an automorphism of G.

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