Here we introduce the implementation of the software package CREP initially designed for MAPLE.
‣ IsUnitForm | ( category ) |
The category for unit forms, which we identify with symmetric integral matrices with 2 along the diagonal.
‣ BilinearFormOfUnitForm( B ) | ( attribute ) |
Arguments: B -- a unit form.
Returns: the bilinear form associated to a unit form B.
The bilinear form associated to the unitform B given by a matrix B is defined for two vectors x and y as: x*B*y^T.
‣ IsWeaklyNonnegativeUnitForm( B ) | ( property ) |
Arguments: B -- a unit form.
Returns: true is the unitform B is weakly non-negative, otherwise false.
The unit form B is weakly non-negative is B(x,y) ≥ 0 for all x≠ 0 in Z^n, where n is the dimension of the square matrix associated to B.
‣ IsWeaklyPositiveUnitForm( B ) | ( property ) |
Arguments: B -- a unit form.
Returns: true is the unitform B is weakly positive, otherwise false.
The unit form B is weakly positive if B(x,y) > 0 for all x≠ 0 in Z^n, where n is the dimension of the square matrix associated to B.
‣ PositiveRootsOfUnitForm( B ) | ( attribute ) |
Arguments: B -- a unit form.
Returns: the 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!".
This attribute will be attached to B when IsWeaklyPositiveUnitForm is applied to B and it is weakly positive.
‣ QuadraticFormOfUnitForm( B ) | ( attribute ) |
Arguments: B -- a unit form.
Returns: the quadratic form associated to a unit form B.
The quadratic form associated to the unitform B given by a matrix B is defined for a vector x as: frac12x*B*x^T.
‣ SymmetricMatrixOfUnitForm( B ) | ( attribute ) |
Arguments: B -- a unit form.
Returns: the symmetric integral matrix which defines the unit form B.
‣ TitsUnitFormOfAlgebra( A ) | ( operation ) |
Arguments: A -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).
Returns: the Tits unit form associated to the algebra A.
This 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 B such that for x = (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, where n is the number of vertices in Q.
‣ EulerBilinearFormOfAlgebra( A ) | ( operation ) |
Arguments: A -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).
Returns: the Euler (non-symmetric) bilinear form associated to the algebra A.
This function returns the Euler (non-symmetric) bilinear form associated to a finite dimensional (basic) quotient of a path algebra A. That is, it returns a bilinear form (function) defined by
f(x,y) = x*CartanMatrix(A)^(-1)*y
It makes sense only in case A is of finite global dimension.
‣ UnitForm( B ) | ( operation ) |
Arguments: B -- an integral matrix.
Returns: the unit form in the category IsUnitForm (12.2-1) associated to the matrix B.
The function checks if B is a symmetric integral matrix with 2 along the diagonal, and returns an error message otherwise. In addition it sets the attributes, BilinearFormOfUnitForm (12.2-2), QuadraticFormOfUnitForm (12.2-6) and SymmetricMatrixOfUnitForm (12.2-7).
‣ CombinatorialMap( n, sigma, iota, m ) | ( operation ) |
Arguments : n -- number of half-edges, sigma -- a permutation specifying the ordering of half-edges aroud vertices, iota -- an involution pairing the half-edges, m -- a list of marked half-edges
Returns: a combinatorial map, which is an object from the category IsCombinatorialMap (???).
The size of the underlying set given by size is given a positive integer, the ordering and the pairing are given by two permutations ordering and paring of a set of size elements ({1, 2,..., size}) and the marked half-edges are given as a list of integers in {1, 2,..., size}.
‣ FacesOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: the list of faces of the combinatorial map map.
The faces are given by a list of half-edges of map.
‣ GenusOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: the genus of the surface represented by the combinatorial map map.
‣ DualOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: the dual combinatorial map of map.
‣ MaximalPathsOfGentleAlgebra( A ) | ( attribute ) |
Argument : A -- a gentle algebra.
Returns: the list of maximal paths of the quiver with relations defining A.
Only returns paths of non-zero length.
‣ RemoveEdgeOfCombinatorialMap( combmap, E ) | ( operation ) |
Argument : map -- a combinatorial map, edge -- a list of two paired half-edges.
Returns: the combinatorial map obtained by removing edge from the combinatorial map map.
The argument edge has to be a list consisting of two half-edges of map 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.
‣ CombinatorialMapOfGentleAlgebra( A ) | ( attribute ) |
Argument : A -- a gentle algebra.
Returns: the combinatorial map corresponding to the Brauer Graph of the trivial extension of algebra with regard to its dual.
The function assumes that algebra is gentle.
‣ MarkedBoundariesOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: a list consisting of pairs [Face,numberofmarkedpoints] where Face is a face of the combinatorial map map and numberofmarkedpoints is the number of marked half-edges on the corresponding boundary component.
‣ WindingNumber( map, gamma ) | ( operation ) |
Argument : map -- a combinatorial map, gamma -- a list of half-edges forming a closed curve on the surface.
Returns: the combinatorial winding number of gamma on the dissected surface given by map.
‣ BoundaryCurvesOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: a list whose elements are lists of half-edges of the combiantorial map map. Each of these lists corresponds to a closed curve homotopic to a boundary component of the surface represented by map.
The 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.
‣ DepthSearchCombinatorialMap( combmap, x ) | ( operation ) |
Argument : map -- a combinatorial map, x -- a half-edge of map
Returns: a list of pairs of paired half-edges of the combinatorial map map corresponding to the cover tree of the underlying graph of map obtained by a depth first search with root the vertex to which is attached x.
The paired half-edges of the cover tree are oriented towards the root of the cover tree.
‣ WidthSearchCombinatorialMap( combmap, x ) | ( operation ) |
Argument : map -- a combinatorial map, x -- a half-edge of map
Returns: a list of pairs of paired half-edges of the combinatorial map map corresponding to the cover tree of the underlying graph of map obtained by a width first search with root the vertex to which is attached x.
The paired half-edges of the cover tree are oriented towards the root of the cover tree.
‣ NonSeperatingCurve( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: a list of half-edges of the combinatorial map map corresponding to a non-seperating closed curve of the surface.
This function must be used on a combinatorial map of genus at least one.
‣ CutNonSepCombinatorialMap( map, alpha, index ) | ( operation ) |
Arguments : map -- a combinatorial map, alpha -- a list of half-edges, index -- a list.
Returns: a triplet [newmap, [boundary1,boundary2], newindex] where newmap is the combinatorial map obtained by cutting the surface represented by map along the closed curve curve. The lists boundary1 and boundary2 are the boundaries of newmap that were created by the cut.
The argument index 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.
‣ JoinCurveCombinatorialMap( map, bound1, bound2, index ) | ( operation ) |
Arguments : map -- a combinatorial map, boundary1 -- a list of half-edges, boundary2 -- a list of half-edges, index -- a list.
Returns: a list corresponding to a simple curve of map joining the two boundaries boundary1 and boundary2 in such a way that using index it corresponds to a closed curve on the original combinatorial map.
The list consists of adjacent half-edges of map which form the closed curve.
‣ CutJoinCurveCombinatorialMap( map, bound1, bound2, beta, index ) | ( operation ) |
Arguments : map -- a combinatorial map, boundary1 -- a list of half-edges, boundary2 -- a list of half-edges, beta -- a list of half-edges corresponding to a curve, index -- a list.
Returns: the pair [newmap, newindex] where newmap is the combinatorial map obtained by cutting the combinatorial map map along curve which has to be a simple curve joining boundary1 and boundary2
‣ HomologyBasisOfCombinatorialMap( map ) | ( attribute ) |
Argument : map -- a combinatorial map.
Returns: a list of pairs [a_i,b_i] where a_i and b_i are lists of half-edges of the combinatorial map map 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 map for the intersection form.
‣ AreDerivedEquivalent( A, B ) | ( operation ) |
Arguments : A -- a gentle algebra, B -- a gentle algebra.
Returns: true if A and B are derived equivalent and false otherwise. The arguments must be gentle algebras over the same field.
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