  
  [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
  
  [33X[1;0Y[29X[2XIsUnitForm[102X [32X Category[133X
  
  [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
  
  [33X[1;0Y[29X[2XBilinearFormOfUnitForm[102X( [3XB[103X ) [32X attribute[133X
  
  [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
  
  [33X[1;0Y[29X[2XIsWeaklyNonnegativeUnitForm[102X( [3XB[103X ) [32X property[133X
  
  [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
  
  [33X[1;0Y[29X[2XIsWeaklyPositiveUnitForm[102X( [3XB[103X ) [32X property[133X
  
  [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
  
  [33X[1;0Y[29X[2XPositiveRootsOfUnitForm[102X( [3XB[103X ) [32X attribute[133X
  
  [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
  
  [33X[1;0Y[29X[2XQuadraticFormOfUnitForm[102X( [3XB[103X ) [32X attribute[133X
  
  [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
  
  [33X[1;0Y[29X[2XSymmetricMatrixOfUnitForm[102X( [3XB[103X ) [32X attribute[133X
  
  [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
  
  [33X[1;0Y[29X[2XTitsUnitFormOfAlgebra[102X( [3XA[103X ) [32X operation[133X
  
  [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
  
  [33X[1;0Y[29X[2XEulerBilinearFormOfAlgebra[102X( [3XA[103X ) [32X operation[133X
  
  [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
  
  [33X[1;0Y[29X[2XUnitForm[102X( [3XB[103X ) [32X operation[133X
  
  [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
  
  
  [1X12.3 [33X[0;0YCombinatorial Maps[133X[101X
  
  [1X12.3-1 IsCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XIsCombinatorialMap[102X(  ) [32X filter[133X
  
  [33X[0;0YThis defines the category [2XIsCombinatorialMap[102X.[133X
  
  [1X12.3-2 CombinatorialMap[101X
  
  [33X[1;0Y[29X[2XCombinatorialMap[102X( [3Xn[103X, [3Xsigma[103X, [3Xiota[103X, [3Xm[103X ) [32X operation[133X
  
  [33X[0;0YArguments : [3Xn[103X -- number of half-edges, [3Xsigma[103X -- a permutation specifying the
  ordering  of  half-edges  aroud  vertices, [3Xiota[103X -- an involution pairing the
  half-edges, [3Xm[103X -- a list of marked half-edges[133X
  
  [6XReturns:[106X  [33X[0;10Ya  combinatorial  map,  which  is  an  object  from  the  category
            [2XIsCombinatorialMap[102X ([14X12.3-1[114X).[133X
  
  [33X[0;0YThe  size  of  the underlying set given by [3Xsize[103X is given a positive integer,
  the  ordering  and  the  pairing  are given by two permutations [3Xordering[103X and
  [3Xparing[103X  of  a  set  of  [3Xsize[103X  elements  ({1,  2,...,  [3Xsize[103X})  and the marked
  half-edges are given as a list of integers in {1, 2,..., [3Xsize[103X}.[133X
  
  [1X12.3-3 FacesOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XFacesOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe list of faces of the combinatorial map [3Xmap[103X.[133X
  
  [33X[0;0YThe faces are given by a list of half-edges of [3Xmap[103X.[133X
  
  [1X12.3-4 GenusOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XGenusOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe genus of the surface represented by the combinatorial map [3Xmap[103X.[133X
  
  [1X12.3-5 DualOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XDualOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe dual combinatorial map of [3Xmap[103X.[133X
  
  [1X12.3-6 MaximalPathsOfGentleAlgebra[101X
  
  [33X[1;0Y[29X[2XMaximalPathsOfGentleAlgebra[102X( [3XA[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3XA[103X -- a gentle algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe list of maximal paths of the quiver with relations defining [3XA[103X.[133X
  
  [33X[0;0YOnly returns paths of non-zero length.[133X
  
  [1X12.3-7 RemoveEdgeOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XRemoveEdgeOfCombinatorialMap[102X( [3Xcombmap[103X, [3XE[103X ) [32X operation[133X
  
  [33X[0;0YArgument  :  [3Xmap[103X  --  a  combinatorial  map,  [3Xedge[103X  --  a list of two paired
  half-edges.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe   combinatorial   map  obtained  by  removing  [3Xedge[103X  from  the
            combinatorial map [3Xmap[103X.[133X
  
  [33X[0;0YThe  argument [3Xedge[103X has to be a list consisting of two half-edges of [3Xmap[103X that
  are  paired.  The  returned  combinatorial map does not change size, the two
  removed  half-edges  are  now  fixed  by  the pairing and ordering so do not
  appear in non-trivial orbits.[133X
  
  [1X12.3-8 CombinatorialMapOfGentleAlgebra[101X
  
  [33X[1;0Y[29X[2XCombinatorialMapOfGentleAlgebra[102X( [3XA[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3XA[103X -- a gentle algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  combinatorial  map  corresponding  to the Brauer Graph of the
            trivial extension of [3Xalgebra[103X with regard to its dual.[133X
  
  [33X[0;0YThe function assumes that [3Xalgebra[103X is gentle.[133X
  
  [1X12.3-9 MarkedBoundariesOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XMarkedBoundariesOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  list consisting of pairs [Face,numberofmarkedpoints] where Face
            is a face of the combinatorial map [3Xmap[103X and numberofmarkedpoints is
            the  number  of  marked  half-edges  on the corresponding boundary
            component.[133X
  
  [1X12.3-10 WindingNumber[101X
  
  [33X[1;0Y[29X[2XWindingNumber[102X( [3Xmap[103X, [3Xgamma[103X ) [32X operation[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map, [3Xgamma[103X -- a list of half-edges forming
  a closed curve on the surface.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe combinatorial winding number of [3Xgamma[103X on the dissected surface
            given by [3Xmap[103X.[133X
  
  [1X12.3-11 BoundaryCurvesOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XBoundaryCurvesOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ya list whose elements are lists of half-edges of the combiantorial
            map  [3Xmap[103X.  Each  of  these  lists  corresponds  to  a closed curve
            homotopic  to  a  boundary component of the surface represented by
            [3Xmap[103X.[133X
  
  [33X[0;0YThe  closed  curves  are  represented  by a list of adjacent half-edges. The
  orientation  of  these  curves  is chosen so that the corresponding boundary
  component  is  to the right. The returned curves correspond to the curves in
  minimal  position  in  regards to the dual dissection of the surface. In the
  case  of  the  disk,  the  boundary  curve  is trivial and the empty list is
  returned.[133X
  
  [1X12.3-12 DepthSearchCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XDepthSearchCombinatorialMap[102X( [3Xcombmap[103X, [3Xx[103X ) [32X operation[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map, [3Xx[103X -- a half-edge of [3Xmap[103X[133X
  
  [6XReturns:[106X  [33X[0;10Ya  list of pairs of paired half-edges of the combinatorial map [3Xmap[103X
            corresponding  to  the  cover  tree of the underlying graph of [3Xmap[103X
            obtained  by a depth first search with root the vertex to which is
            attached [3Xx[103X.[133X
  
  [33X[0;0YThe paired half-edges of the cover tree are oriented towards the root of the
  cover tree.[133X
  
  [1X12.3-13 WidthSearchCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XWidthSearchCombinatorialMap[102X( [3Xcombmap[103X, [3Xx[103X ) [32X operation[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map, [3Xx[103X -- a half-edge of [3Xmap[103X[133X
  
  [6XReturns:[106X  [33X[0;10Ya  list of pairs of paired half-edges of the combinatorial map [3Xmap[103X
            corresponding  to  the  cover  tree of the underlying graph of [3Xmap[103X
            obtained  by a width first search with root the vertex to which is
            attached [3Xx[103X.[133X
  
  [33X[0;0YThe paired half-edges of the cover tree are oriented towards the root of the
  cover tree.[133X
  
  [1X12.3-14 NonSeperatingCurve[101X
  
  [33X[1;0Y[29X[2XNonSeperatingCurve[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ya list of half-edges of the combinatorial map [3Xmap[103X corresponding to
            a non-seperating closed curve of the surface.[133X
  
  [33X[0;0YThis function must be used on a combinatorial map of genus at least one.[133X
  
  [1X12.3-15 CutNonSepCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XCutNonSepCombinatorialMap[102X( [3Xmap[103X, [3Xalpha[103X, [3Xindex[103X ) [32X operation[133X
  
  [33X[0;0YArguments : [3Xmap[103X -- a combinatorial map, [3Xalpha[103X -- a list of half-edges, [3Xindex[103X
  -- a list.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  triplet  [newmap, [boundary1,boundary2], newindex] where newmap
            is   the   combinatorial  map  obtained  by  cutting  the  surface
            represented  by  [3Xmap[103X  along  the  closed  curve  [3Xcurve[103X.  The lists
            boundary1  and  boundary2  are  the boundaries of newmap that were
            created by the cut.[133X
  
  [33X[0;0YThe  argument [3Xindex[103X is a list whose i-th element is the half-edge to which i
  corresponds  in  some original combinatorial map. The result newindex is the
  updated index for newmap where all created half-edges are added.[133X
  
  [1X12.3-16 JoinCurveCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XJoinCurveCombinatorialMap[102X( [3Xmap[103X, [3Xbound1[103X, [3Xbound2[103X, [3Xindex[103X ) [32X operation[133X
  
  [33X[0;0YArguments  :  [3Xmap[103X -- a combinatorial map, [3Xboundary1[103X -- a list of half-edges,
  [3Xboundary2[103X -- a list of half-edges, [3Xindex[103X -- a list.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  list  corresponding  to  a  simple curve of [3Xmap[103X joining the two
            boundaries  [3Xboundary1[103X and [3Xboundary2[103X in such a way that using [3Xindex[103X
            it  corresponds  to  a  closed curve on the original combinatorial
            map.[133X
  
  [33X[0;0YThe list consists of adjacent half-edges of [3Xmap[103X which form the closed curve.[133X
  
  [1X12.3-17 CutJoinCurveCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XCutJoinCurveCombinatorialMap[102X( [3Xmap[103X, [3Xbound1[103X, [3Xbound2[103X, [3Xbeta[103X, [3Xindex[103X ) [32X operation[133X
  
  [33X[0;0YArguments  :  [3Xmap[103X -- a combinatorial map, [3Xboundary1[103X -- a list of half-edges,
  [3Xboundary2[103X   --   a  list  of  half-edges,  [3Xbeta[103X  --  a  list  of  half-edges
  corresponding to a curve, [3Xindex[103X -- a list.[133X
  
  [6XReturns:[106X  [33X[0;10Ythe  pair [newmap, newindex] where newmap is the combinatorial map
            obtained  by  cutting  the combinatorial map [3Xmap[103X along [3Xcurve[103X which
            has to be a simple curve joining [3Xboundary1[103X and [3Xboundary2[103X[133X
  
  [1X12.3-18 HomologyBasisOfCombinatorialMap[101X
  
  [33X[1;0Y[29X[2XHomologyBasisOfCombinatorialMap[102X( [3Xmap[103X ) [32X attribute[133X
  
  [33X[0;0YArgument : [3Xmap[103X -- a combinatorial map.[133X
  
  [6XReturns:[106X  [33X[0;10Ya  list  of  pairs  [a_i,b_i]  where  a_i  and  b_i  are  lists of
            half-edges  of  the  combinatorial  map [3Xmap[103X forming a closed curve
            such  that  the  a_i  and b_i form a simplectic basis of the first
            homology   group  of  the  surface  represented  by  [3Xmap[103X  for  the
            intersection form.[133X
  
  [1X12.3-19 AreDerivedEquivalent[101X
  
  [33X[1;0Y[29X[2XAreDerivedEquivalent[102X( [3XA[103X, [3XB[103X ) [32X operation[133X
  
  [33X[0;0YArguments : [3XA[103X -- a gentle algebra, [3XB[103X -- a gentle algebra.[133X
  
  [6XReturns:[106X  [33X[0;10Ytrue  if  [3XA[103X  and [3XB[103X are derived equivalent and false otherwise. The
            arguments must be gentle algebras over the same field.[133X
  
