  
  [1X5 [33X[0;0YFunctions for testing Majorana representations[133X[101X
  
  [33X[0;0YThe  output  of the function [2XMajoranaRepresentation[102X ([14X3.1-1[114X) 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[133X
  
  [30X    [33X[0;6Ythe  inner product is a Frobenius form (see [2XMAJORANA_TestFrobeniusForm[102X
        ([14X5.2-1[114X));[133X
  
  [30X    [33X[0;6Ythe  inner product is positive definite (see [2XMAJORANA_TestInnerProduct[102X
        ([14X5.2-2[114X));[133X
  
  [30X    [33X[0;6Ythe   inner   product   obeys  axiom  M2  (Norton's  inequality)  (see
        [2XMAJORANA_TestAxiomM2[102X ([14X5.2-3[114X));[133X
  
  [30X    [33X[0;6Ythe algebra is primitive (see [2XMAJORANA_TestPrimitivity[102X ([14X5.2-4[114X)).[133X
  
  
  [1X5.1 [33X[0;0YThe main function[133X[101X
  
  [1X5.1-1 MajoranaAlgebraTest[101X
  
  [33X[1;0Y[29X[2XMajoranaAlgebraTest[102X( [3Xrep[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the algebra given by [3Xrep[103X is indeed a Majorana algebra.[133X
  
  [33X[0;0YNote:  does not check that the algebra obeys axiom M2 (Norton's inequality),
  this can be separately tested using [2XMAJORANA_TestAxiomM2[102X ([14X5.2-3[114X).[133X
  
  
  [1X5.2 [33X[0;0YOther functions[133X[101X
  
  [1X5.2-1 MAJORANA_TestFrobeniusForm[101X
  
  [33X[1;0Y[29X[2XMAJORANA_TestFrobeniusForm[102X( [3Xrep[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X  if  the  inner  product  given  by  [3Xrep.innerproducts[103X  is  a
            Frobenius form, otherwise returns false.[133X
  
  [1X5.2-2 MAJORANA_TestInnerProduct[101X
  
  [33X[1;0Y[29X[2XMAJORANA_TestInnerProduct[102X( [3Xrep[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X  if  the inner product given by [3Xrep.innerproducts[103X is positive
            definite, otherwise returns false.[133X
  
  [1X5.2-3 MAJORANA_TestAxiomM2[101X
  
  [33X[1;0Y[29X[2XMAJORANA_TestAxiomM2[102X( [3Xrep[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X  if  the inner product given by [3Xrep.innerproducts[103X obeys axiom
            M2 (Norton's inequality), otherwise returns false.[133X
  
  [1X5.2-4 MAJORANA_TestPrimitivity[101X
  
  [33X[1;0Y[29X[2XMAJORANA_TestPrimitivity[102X( [3Xrep[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[3Xtrue[103X if the 1-eigenspaces of all axes are 1-dimensional, otherwise
            returns false.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlternatingGroup(5);;[127X[104X
    [4X[25Xgap>[125X [27XT := AsList( ConjugacyClass(G, (1,2)(3,4)));;[127X[104X
    [4X[25Xgap>[125X [27Xinput := ShapesOfMajoranaRepresentation(G,T);;[127X[104X
    [4X[25Xgap>[125X [27Xrep := MajoranaRepresentation(input, 2);;[127X[104X
    [4X[25Xgap>[125X [27XNClosedMajoranaRepresentation(rep);;[127X[104X
    [4X[25Xgap>[125X [27XMAJORANA_IsComplete(rep);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMajoranaAlgebraTest(rep);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMAJORANA_TestFrobeniusForm(rep);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMAJORANA_TestInnerProduct(rep);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMAJORANA_TestAxiomM2(rep);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMAJORANA_TestPrimitivity(rep);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
