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### 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`.