`‣ MajoranaRepresentation` ( input, index[, options] ) | ( function ) |

Returns: a record giving a Majorana representation

This takes two or three arguments, the first of which must be the output of the function `ShapesOfMajoranaRepresentation`

(2.1-1) and the second of which is the index of the desired shape in list `input.shapes`.

If the optional argument `options` is given then it must be a record. The following components of `options` are recognised:

`axioms`

This component must be bound to the string

`"AllAxioms"`or`"NoAxioms"`. If bound to`"AllAxioms"`then the algorithm assumes the axioms 2Aa, 2Ab, 3A, 4A and 5A as in Seress (2012). If bound to`"NoAxioms"`then the algorithm only assumes the Majorana axioms M1 - M7. The default value is`"AllAxioms"`.`form`

If this is bound to

`true`then the algorithm assume the existence of an inner product (as in the definition of a Majorana algebra). Otherwise, if bound to`false`then no inner product is assumed (and we are in fact constructing an axial algebra that satisfies the Majorana fusion law). The default value is`true`.`embedding`

If this is bound to

`true`then the algorithm first attempts to construct large subalgebras of the final representation before starting the main construction. The default value is`false`.

A Majorana algebra \(V\) generated by a set of axes \(A\) is called \(n\)-closed if it is spanned as a vector space by products of elements of \(A\) of length at most \(n\). As most known Majorana algebras are \(2\)-closed, the function `MajoranaRepresentation`

(3.1-1) only attempts to construct the \(2\)-closed part.

If it is not successful then the output is a partial Majorana representation, i.e. a Majorana representation with some missing algebra products. In this case, the function `MAJORANA_IsComplete`

(4.2-1) returns false.

If the user wishes, they may then pass this incomplete Majorana representation to the function `NClosedMajoranaRepresentation`

(3.2-1) in order to attempt construction of the \(3\)-closed part. This process may then be repeated as many times as the user wishes.

`‣ NClosedMajoranaRepresentation` ( rep ) | ( function ) |

Takes as its input an incomplete Majorana representation rep that has been generated using the function `MajoranaRepresentation`

(3.1-1). Again runs the main algorithm in order to attempt construction of the \(3\)-closed part of the algebra. If the function `NClosedMajoranaRepresentation`

is called \(n\) times on the same Majorana representation rep then this representation will be the \(n + 2\)-closed part of the algebra.

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