  
  [1X12 [33X[0;0YCombinatorial representation theory[133X[101X
  
  
  [1X12.1 [33X[0;0YIntroduction[133X[101X
  
  [33X[0;0YHere  we introduce the implementation of the software package CREP initially
  designed for MAPLE.[133X
  
  
  [1X12.2 [33X[0;0YDifferent unit forms[133X[101X
  
  [1X12.2-1 IsUnitForm[101X
  
  [29X[2XIsUnitForm[102X[32X Category
  
  [33X[0;0YThe  category  for  unit  forms,  which  we identify with symmetric integral
  matrices with 2 along the diagonal.[133X
  
  [1X12.2-2 BilinearFormOfUnitForm[101X
  
  [29X[2XBilinearFormOfUnitForm[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe bilinear form associated to a unit form [3XB[103X.[133X
  
  [33X[0;0YThe  bilinear  form  associated  to  the  unitform  [3XB[103X given by a matrix [10XB[110X is
  defined for two vectors [10Xx[110X and [10Xy[110X as: [22Xx*B*y^T[122X.[133X
  
  [1X12.2-3 IsWeaklyNonnegativeUnitForm[101X
  
  [29X[2XIsWeaklyNonnegativeUnitForm[102X( [3XB[103X ) [32X property
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue is the unitform [3XB[103X is weakly non-negative, otherwise false.[133X
  
  [33X[0;0YThe  unit  form  [3XB[103X is weakly non-negative is [22XB(x,y) ≥ 0[122X for all [22Xx≠ 0[122X in [22XZ^n[122X,
  where [22Xn[122X is the dimension of the square matrix associated to [3XB[103X.[133X
  
  [1X12.2-4 IsWeaklyPositiveUnitForm[101X
  
  [29X[2XIsWeaklyPositiveUnitForm[102X( [3XB[103X ) [32X property
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue is the unitform [3XB[103X is weakly positive, otherwise false.[133X
  
  [33X[0;0YThe  unit form [3XB[103X is weakly positive if [22XB(x,y) > 0[122X for all [22Xx≠ 0[122X in [22XZ^n[122X, where
  [22Xn[122X is the dimension of the square matrix associated to [3XB[103X.[133X
  
  [1X12.2-5 PositiveRootsOfUnitForm[101X
  
  [29X[2XPositiveRootsOfUnitForm[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  positive  roots  of  a  unit form, if the unit form is weakly
            positive. If they have not been computed, an error message will be
            returned saying "no method found!".[133X
  
  [33X[0;0YThis  attribute  will  be  attached  to  [3XB[103X  when [10XIsWeaklyPositiveUnitForm[110X is
  applied to [3XB[103X and it is weakly positive.[133X
  
  [1X12.2-6 QuadraticFormOfUnitForm[101X
  
  [29X[2XQuadraticFormOfUnitForm[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe quadratic form associated to a unit form [3XB[103X.[133X
  
  [33X[0;0YThe  quadratic  form  associated  to  the  unitform [3XB[103X given by a matrix [10XB[110X is
  defined for a vector [10Xx[110X as: [22Xfrac12x*B*x^T[122X.[133X
  
  [1X12.2-7 SymmetricMatrixOfUnitForm[101X
  
  [29X[2XSymmetricMatrixOfUnitForm[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YArguments: [3XB[103X -- a unit form.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe symmetric integral matrix which defines the unit form [3XB[103X.[133X
  
  [1X12.2-8 TitsUnitFormOfAlgebra[101X
  
  [29X[2XTitsUnitFormOfAlgebra[102X( [3XA[103X ) [32X operation
  
  [33X[0;0YArguments:  [3XA[103X  --  a  finite dimensional (quotient of a) path algebra (by an
  admissible ideal).[133X
  
  [6XReturns:[106X  [33X[0;10Ythe Tits unit form associated to the algebra [3XA[103X.[133X
  
  [33X[0;0YThis  function  returns the Tits unitform associated to a finite dimensional
  quotient  of  a  path  algebra by an admissible ideal or path algebra, given
  that the underlying quiver has no loops or minimal relations that starts and
  ends  in the same vertex. That is, then it returns a symmetric matrix [22XB[122X such
  that  for  [22Xx  =  (x_1,...,x_n) (1/2)*(x_1,...,x_n)B(x_1,...,x_n)^T = ∑_i=1^n
  x_i^2 - ∑_i,j dim_k Ext^1(S_i,S_j)x_ix_j + ∑_i,j dim_k Ext^2(S_i,S_j)x_ix_j[122X,
  where [22Xn[122X is the number of vertices in [22XQ[122X.[133X
  
  [1X12.2-9 EulerBilinearFormOfAlgebra[101X
  
  [29X[2XEulerBilinearFormOfAlgebra[102X( [3XA[103X ) [32X operation
  
  [33X[0;0YArguments:  [3XA[103X  --  a  finite dimensional (quotient of a) path algebra (by an
  admissible ideal).[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  Euler (non-symmetric) bilinear form associated to the algebra
            [3XA[103X.[133X
  
  [33X[0;0YThis  function returns the Euler (non-symmetric) bilinear form associated to
  a  finite  dimensional  (basic)  quotient  of  a path algebra [3XA[103X. That is, it
  returns a bilinear form (function) defined by[133X
  [33X[0;0Y[22Xf(x,y) = x*CartanMatrix(A)^(-1)*y[122X[133X
  [33X[0;0YIt makes sense only in case [3XA[103X is of finite global dimension.[133X
  
  [1X12.2-10 UnitForm[101X
  
  [29X[2XUnitForm[102X( [3XB[103X ) [32X operation
  
  [33X[0;0YArguments: [3XB[103X -- an integral matrix.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  unit  form  in the category [2XIsUnitForm[102X ([14X12.2-1[114X) associated to
            the matrix [3XB[103X.[133X
  
  [33X[0;0YThe  function  checks  if  [3XB[103X is a symmetric integral matrix with 2 along the
  diagonal,  and  returns  an error message otherwise. In addition it sets the
  attributes,    [2XBilinearFormOfUnitForm[102X    ([14X12.2-2[114X),   [2XQuadraticFormOfUnitForm[102X
  ([14X12.2-6[114X) and [2XSymmetricMatrixOfUnitForm[102X ([14X12.2-7[114X).[133X
  
