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### 3 Majorana representations

#### 3.1 The main function

##### 3.1-1 MajoranaRepresentation
 ‣ 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.

#### 3.2 The n-closed function

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.

##### 3.2-1 NClosedMajoranaRepresentation
 ‣ 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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