GAP (MajoranaAlgebras) - Chapter 5: Functions for testing Majorana representations

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5 Functions for testing Majorana representations
5.1 The main function
5.1-1 MajoranaAlgebraTest
5.2 Other functions
5.2-1 MAJORANA_TestFrobeniusForm
5.2-2 MAJORANA_TestInnerProduct
5.2-3 MAJORANA_TestAxiomM2
5.2-4 MAJORANA_TestPrimitivity

### 5 Functions for testing Majorana representations

The output of the function `MajoranaRepresentation` (3.1-1) 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

- the inner product is a Frobenius form (see `MAJORANA_TestFrobeniusForm` (5.2-1));

- the inner product is positive definite (see `MAJORANA_TestInnerProduct` (5.2-2));

- the inner product obeys axiom M2 (Norton's inequality) (see `MAJORANA_TestAxiomM2` (5.2-3));

- the algebra is primitive (see `MAJORANA_TestPrimitivity` (5.2-4)).

#### 5.1 The main function

##### 5.1-1 MajoranaAlgebraTest

`‣ MajoranaAlgebraTest`( rep ) | ( function )

Returns: true if the algebra given by rep is indeed a Majorana algebra.

Note: does not check that the algebra obeys axiom M2 (Norton's inequality), this can be separately tested using `MAJORANA_TestAxiomM2` (5.2-3).

#### 5.2 Other functions

##### 5.2-1 MAJORANA_TestFrobeniusForm

`‣ MAJORANA_TestFrobeniusForm`( rep ) | ( function )

Returns: true if the inner product given by rep.innerproducts is a Frobenius form, otherwise returns false.

##### 5.2-2 MAJORANA_TestInnerProduct

`‣ MAJORANA_TestInnerProduct`( rep ) | ( function )

Returns: true if the inner product given by rep.innerproducts is positive definite, otherwise returns false.

##### 5.2-3 MAJORANA_TestAxiomM2

`‣ MAJORANA_TestAxiomM2`( rep ) | ( function )

Returns: true if the inner product given by rep.innerproducts obeys axiom M2 (Norton's inequality), otherwise returns false.

##### 5.2-4 MAJORANA_TestPrimitivity

`‣ MAJORANA_TestPrimitivity`( rep ) | ( function )

Returns: true if the 1-eigenspaces of all axes are 1-dimensional, otherwise returns false.

    gap> G := AlternatingGroup(5);;
    gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));;
    gap> input := ShapesOfMajoranaRepresentation(G,T);;
    gap> rep := MajoranaRepresentation(input, 2);;
    gap> NClosedMajoranaRepresentation(rep);;
    gap> MAJORANA_IsComplete(rep);
    true
    gap> MajoranaAlgebraTest(rep);
    true
    gap> MAJORANA_TestFrobeniusForm(rep);
    true
    gap> MAJORANA_TestInnerProduct(rep);
    true
    gap> MAJORANA_TestAxiomM2(rep);
    true
    gap> MAJORANA_TestPrimitivity(rep);
    true

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