In order to construct the Majorana representation of a group `G` with respect to a set of involutions `T`, you must first call `ShapesOfMajoranaRepresentation`

(2.1-1).

gap> G := AlternatingGroup(5);; gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));; gap> input := ShapesOfMajoranaRepresentation(G,T);;

This function outputs a record. One component of this record is labelled `shapes` and contains the possible shapes of a Majorana representation of the form \((G,T,V)\).

gap> input.shapes; [ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ], [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ]

To construct the Majorana representation with shape at position `i` of this list, call the function `MajoranaRepresentation`

(3.1-1) with `input` as its first argument and `i` as its second.

gap> rep := MajoranaRepresentation(input, 1);; gap> rep.shape; [ "1A", "2B", "5A", "3C", "5A" ]

There are then a number of functions (see 4) that one case use on the (potentially incomplete) Majorana representation that this function has outputted.

gap> MAJORANA_IsComplete(rep); true gap> MAJORANA_Dimension(rep); 21

If an incomplete algebra is returned then the function `NClosedMajoranaRepresentation`

(3.2-1) can be used to attempt to find the 3-closed part of the algebra.

gap> G := AlternatingGroup(5);; gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));; gap> input := ShapesOfMajoranaRepresentation(G,T);; gap> input.shapes; [ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ], [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ] gap> rep := MajoranaRepresentation(input, 2);; gap> MAJORANA_IsComplete(rep); false gap> NClosedMajoranaRepresentation(rep);; gap> MAJORANA_IsComplete(rep); true gap> MAJORANA_Dimension(rep); 46

*Note that all vectors and matrices are given in sparse matrix format, as provided by the GAP package Gauss. If mat is such a matrix then the integers in mat!.indices refer to a spanning set of the algebra indexed by the list rep.setup.coords. The list mat!.entries give their corresponding coefficients.*

The function `MajoranaRepresentation`

(3.1-1) outputs a record that encodes the information required to perform calculations in the Majorana representation that has been calculated. The record contains the following components.

`group`

The group

`G`, as inputted by the user.`involutions`

The set

`T`, as inputted by the user.`shape`

The shape of the representation, as chosen by the user in the input of

`MajoranaRepresentation`

(3.1-1).`eigenvalues`

A list whose values give the eigenvalues of the adjoint action of the axes of the algebra. In this case, it must be equal to (or a subset of)

`[0, 1/4, 1/32]`. Note that we omit the eigenvalue 1 as we assume the algebra to be primitive.`axioms`

A string representing the axiomatic setting of the algebra's construction, potentially chosen by the user with the

`options`record in the input of`MajoranaRepresentation`

(3.1-1).`setup`

Is itself a record, containing (among others) the following components.

`coords`

A list whose elements index a spanning set of the algebra.

`nullspace`

Again a record such that

`nullspace.vectors`gives a basis of the nullspace of the algebra (as the elements`rep.setup.coords`are not necessarily linearly independent).`orbitreps`

A list of indices giving the representatives of the orbits of the action of the group

`G`on`T`.`pairreps`

A list of pairs of indices giving representatives of the orbitals of the action of the group

`G`on`rep.setup.coords`.

`algebraproducts`

A list where the vector at position

`i`denotes the algebra product of the two spanning set vectors whose indices (in`rep.setup.coords`) are given by`rep.setup.pairreps[i]`. If the`i`th entry is set to`false`then this algebra product has not yet been found and the algebra is incomplete.`innerproducts`

Performs the same role as

`algebraproducts`except that, instead of vectors, the entries are rational numbers denoting the inner product between two spanning set vectors.`evecs`

A list where if

`i`is contained in`rep.setup.orbitreps`then`rep.evecs[i]`is bound to a record. This record has components`"ev"`where`ev`is an eigenvalue contained in`rep.eigenvalues`. This component gives a basis for the eigenspace of the axis corresponding to`rep.involutions[i]`with eigenvalue`ev`.

`‣ InfoMajorana` | ( info class ) |

The default info level of `InfoMajorana` is 0. No information is printed at this level. If the info level is at least 10 then `Success` is printed if the algorithm has produced a complete Majorana algebra, otherwise `Fail` is printed. If the info level is at least 20 then more information is printed about the progress of the algorithm, up to a maximum info level of 100.

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