Functions for testing Majorana representations

The output of the function 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 ); the inner product is positive definite (see ); the inner product obeys axiom M2 (Norton's inequality) (see ); the algebra is primitive (see ).

The main function 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 .
Other functions true if the inner product given by rep.innerproducts is a Frobenius form, otherwise returns false.

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

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

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