@Chapter Introduction
@Section A quick guide
In order to construct the Majorana representation of a group <A>G</A> with respect
to a set of involutions <A>T</A>, you must first call
<Ref Func="ShapesOfMajoranaRepresentation"/>.
@BeginExample
G := AlternatingGroup(5);;
T := AsList( ConjugacyClass(G, (1,2)(3,4)));;
input := ShapesOfMajoranaRepresentation(G,T);;
@EndExample
This function outputs a record. One component of this record is labelled <A>shapes</A>
and contains the possible shapes of a Majorana representation of the form $(G,T,V)$.
@BeginExample
input.shapes;
#! [ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ],
#! [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ]
@EndExample
To construct the Majorana representation with shape at position <A>i</A> of this list,
call the function <Ref Func="MajoranaRepresentation"/> with <A>input</A> as its first argument
and <A>i</A> as its second.
@BeginExample
rep := MajoranaRepresentation(input, 1);;
rep.shape;
#! [ "1A", "2B", "5A", "3C", "5A" ]
@EndExample
There are then a number of functions (see <Ref Chap="Chapter_funcs"/>) that one case use
on the (potentially incomplete) Majorana representation that this function has outputted.
@BeginExample
MAJORANA_IsComplete(rep);
#! true
MAJORANA_Dimension(rep);
#! 21
@EndExample
If an incomplete algebra is returned then the function
<Ref Func="NClosedMajoranaRepresentation"/> can be used to attempt to find the
3-closed part of the algebra.
@BeginExample
G := AlternatingGroup(5);;
T := AsList( ConjugacyClass(G, (1,2)(3,4)));;
input := ShapesOfMajoranaRepresentation(G,T);;
input.shapes;
#! [ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ],
#! [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ]
rep := MajoranaRepresentation(input, 2);;
MAJORANA_IsComplete(rep);
#! false
NClosedMajoranaRepresentation(rep);;
MAJORANA_IsComplete(rep);
#! true
MAJORANA_Dimension(rep);
#! 46
@EndExample
@Section Understanding the output
**Note that all vectors and matrices are given in sparse matrix format, as provided
by the GAP package <A>Gauss</A>. If <A>mat</A> is such a matrix then the integers in
<A>mat!.indices</A> refer to a spanning set of the algebra indexed by the list
<A>rep.setup.coords</A>. The list <A>mat!.entries</A> give their corresponding coefficients.**
The function <Ref Func="MajoranaRepresentation"/> 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.
<List>
<Mark> <C>group</C></Mark>
<Item> The group <A>G</A>, as inputted by the user.</Item>
<Mark> <C>involutions</C></Mark>
<Item> The set <A>T</A>, as inputted by the user.</Item>
<Mark> <C>shape</C></Mark>
<Item> The shape of the representation, as chosen by the user in the input
of <Ref Func="MajoranaRepresentation"/>.</Item>
<Mark> <C>eigenvalues</C></Mark>
<Item> 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)
<A>[0, 1/4, 1/32]</A>. Note that we omit the eigenvalue 1 as we assume the algebra
to be primitive.</Item>
<Mark> <C>axioms</C></Mark>
<Item> A string representing the axiomatic setting of the algebra's construction,
potentially chosen by the user with the <A>options</A> record in the input of
<Ref Func="MajoranaRepresentation"/>.</Item>
<Mark> <C>setup</C></Mark>
<Item> Is itself a record, containing (among others) the following components.
<List>
<Mark> <C>coords</C></Mark>
<Item> A list whose elements index a spanning set of the algebra.</Item>
<Mark> <C>nullspace</C></Mark>
<Item> Again a record such that <A>nullspace.vectors</A> gives a basis
of the nullspace of the algebra (as the elements <A>rep.setup.coords</A>
are not necessarily linearly independent).</Item>
<Mark> <C>orbitreps</C></Mark>
<Item> A list of indices giving the representatives of the orbits of
the action of the group <A>G</A> on <A>T</A>.</Item>
<Mark> <C>pairreps</C></Mark>
<Item> A list of pairs of indices giving representatives of the orbitals
of the action of the group <A>G</A> on <A>rep.setup.coords</A>.</Item>
</List>
</Item>
<Mark> <C>algebraproducts</C></Mark>
<Item> A list where the vector at position <A>i</A> denotes the algebra product
of the two spanning set vectors whose indices (in <A>rep.setup.coords</A>) are
given by <A>rep.setup.pairreps[i]</A>. If the <A>i</A>th entry is set to <A>false</A>
then this algebra product has not yet been found and the algebra is incomplete.</Item>
<Mark> <C>innerproducts</C></Mark>
<Item> Performs the same role as <A>algebraproducts</A> except that, instead of vectors,
the entries are rational numbers denoting the inner product between two spanning set
vectors.</Item>
<Mark> <C>evecs</C></Mark>
<Item> A list where if <A>i</A> is contained in <A>rep.setup.orbitreps</A> then
<A>rep.evecs[i]</A> is bound to a record. This record has components <A>"ev"</A>
where <A>ev</A> is an eigenvalue contained in <A>rep.eigenvalues</A>. This component gives a basis fo
r the eigenspace of the axis corresponding to
<A>rep.involutions[i]</A> with eigenvalue <A>ev</A>.</Item>
</List>
@Section Info levels
@Chapter Shapes of a Majorana representation
@Section The shapes functions
@Chapter Majorana representations
@Section The main function
@Section The n-closed function
@Chapter Functions for calculating with Majorana representations
@ChapterLabel funcs
@Section Calculating products
@Section Basic functions
@Section The subalgebra structure
@Chapter Functions for testing Majorana representations
The output of the function <Ref Func="MajoranaRepresentation"/> is guaranteed to be a commutative algebra generated by idempotents whose eigenspaces obey the Majorana fusion law. To check that the output is truly a Majorana algebra, one must also check that
<List>
<Item> the inner product is a Frobenius form (see <Ref Func="MAJORANA_TestFrobeniusForm"/>); </Item>
<Item> the inner product is positive definite (see <Ref Func="MAJORANA_TestInnerProduct"/>); </Item>
<Item> the inner product obeys axiom M2 (Norton's inequality) (see <Ref Func="MAJORANA_TestAxiomM2"/>); </Item>
<Item> the algebra is primitive (see <Ref Func="MAJORANA_TestPrimitivity"/>). </Item>
</List>
@Section The main function
@Section Other functions
