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### 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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