`‣ MAJORANA_AlgebraProduct` ( u, v, algebraproducts, setup ) | ( function ) |

Returns: the algebra product of vectors `u` and `v`

The arguments `u` and `v` must be row vectors in sparse matrix format. The arguments `algebraproducts` and `setup` must be the components with these names of a representation as outputted by `MajoranaRepresentation`

(3.1-1). The output is the algebra product of `u` and `v`, also in sparse matrix format.

`‣ MAJORANA_InnerProduct` ( u, v, innerproducts, setup ) | ( function ) |

Returns: the inner product of vectors `u` and `v`

The arguments `u` and `v` must be row vectors in sparse matrix format. The arguments `innerproducts` and `setup` must be the components with these names of a representation as outputted by `MajoranaRepresentation`

(3.1-1). The output is the inner product of `u` and `v`.

gap> G := AlternatingGroup(5);; gap> T := AsList(ConjugacyClass(G, (1,2)(3,4)));; gap> input := ShapesOfMajoranaRepresentation(G,T);; gap> rep := MajoranaRepresentation(input, 1);; gap> Size(rep.setup.coords); 21 gap> u := SparseMatrix( 1, 21, [ [ 1 ] ], [ [ 1 ] ], Rationals);; gap> v := SparseMatrix( 1, 21, [ [ 17 ] ], [ [ 1 ] ], Rationals);; gap> MAJORANA_AlgebraProduct(u, v, rep.algebraproducts, rep.setup); <a 1 x 21 sparse matrix over Rationals> gap> MAJORANA_InnerProduct(u, v, rep.innerproducts, rep.setup); -1/8192

`‣ MAJORANA_IsComplete` ( rep ) | ( function ) |

Returns: true is all algebra products have been found, otherwise returns false

Takes a Majorana representation `rep`, as outputted by `MajoranaRepresentation`

(3.1-1). If the representation is complete, that is to say, if the vector space spanned by the basis vectors indexed by the elements in `rep.setup.coords` is closed under the algebra product given by `rep.algebraproducts`, return true. Otherwise, if some products are not known then return false.

`‣ MAJORANA_Dimension` ( rep ) | ( function ) |

Returns: the dimension of the representation `rep` as an integer

Takes a Majorana representation `rep`, as outputted by `MajoranaRepresentation`

(3.1-1) and returns its dimension as a vector space. If the representation is not complete (cf. `MAJORANA_IsComplete`

(4.2-1) ) then this value might not be the true dimension of the algebra.

`‣ MAJORANA_Eigenvectors` ( index, eval, rep ) | ( function ) |

Returns: a basis of the eigenspace of the axis as position `index` with eigenvalue `eval` as a sparse matrix

`‣ MAJORANA_Basis` ( rep ) | ( function ) |

Returns: a sparse matrix that gives a basis of the algebra

`‣ MAJORANA_AdjointAction` ( axis, basis, rep ) | ( function ) |

Returns: a sparse matrix representing the adjoint action of `axis` on `basis`

Takes a Majorana representation `rep`, as outputted by `MajoranaRepresentation`

(3.1-1), a row vector `axis` in sparse matrix format and a set of basis vectors, also in sparse matrix format. Returns a matrix, also in sparse matrix format, that represents the adjoint action of `axis` on `basis`.

`‣ MAJORANA_Subalgebra` ( vecs, rep ) | ( function ) |

Returns: the subalgebra of the representation `rep` that is generated by `vecs`

Takes a Majorana representation `rep`, as outputted by `MajoranaRepresentation`

(3.1-1) and a set of vectors `vecs` in sparse matrix format and returns the subalgebra generated by `vecs`, also in sparse matrix format.

`‣ MAJORANA_IsJordanAlgebra` ( subalg, rep ) | ( function ) |

Returns: true if the subalgebra `subalg` is a Jordan algebra, otherwise returns false

Takes a Majorana representation `rep`, as outputted by `MajoranaRepresentation`

(3.1-1) and a subalgebra `subalg` of rep. If this subalgebra is a Jordan algebra then function returns true, otherwise returns false.

gap> G := G := AlternatingGroup(5);; gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));; gap> input := ShapesOfMajoranaRepresentation(G,T);; gap> rep := MajoranaRepresentation(input, 2);; gap> MAJORANA_IsComplete(rep); false gap> NClosedMajoranaRepresentation(rep);; gap> MAJORANA_IsComplete(rep); true gap> MAJORANA_Dimension(rep); 46 gap> basis := MAJORANA_Basis(rep); <a 46 x 61 sparse matrix over Rationals> gap> subalg := MAJORANA_Subalgebra(basis, rep); <a 46 x 61 sparse matrix over Rationals> gap> MAJORANA_IsJordanAlgebra(subalg, rep); false

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