‣ IsQuiverRepresentationElement( arg ) | ( filter ) |
Returns: true or false
Category for elements of a quiver representation.
‣ IsQuiverRepresentation( arg ) | ( filter ) |
Returns: true or false
Category for quiver representations.
‣ QuiverRepresentationElement( R, vectors ) | ( operation ) |
‣ QuiverRepresentationElementNC( R, vectors ) | ( operation ) |
Returns: IsQuiverRepresentationElement (3.1-1)
Construct the element in the quiver representation R given by the vectors in the list vectors. Each entry in vectors corresponds to a vertex in the quiver.
The variant QuiverRepresentationElementNC does not check that the contents of the list vectors is appropriate for an element in the representation R.
‣ QuiverRepresentationElementByVertices( R, vertices, vectors ) | ( operation ) |
Returns: IsQuiverRepresentationElement (3.1-1)
Construct an element in the representation R by assigning vectors to certain given vertices. All vertices of the quiver which are not present in the list vertices are assigned the zero vector.
‣ RepresentationOfElement( e ) | ( attribute ) |
Returns: IsQuiverRepresentation (3.1-2)
Returns the quiver representation which the element e belongs to.
‣ ElementVectors( e ) | ( attribute ) |
Returns: IsDenseList (Reference: IsDenseList)
Returns all the vectors of the element e, as a list ordered in the same way as the vertices of the quiver.
‣ ElementVector( e, i ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the vector of the element e at the vertex with number i.
‣ ElementVector( e, v ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the vector of the element e at the vertex v.
‣ PathAction( e, p ) | ( operation ) |
Returns: IsQuiverRepresentationElement (3.1-1)
Produces the element which is the result of letting the path p act on the element e.
‣ QuiverAlgebraAction( re, ae ) | ( operation ) |
Returns: IsQuiverRepresentationElement (3.1-1)
Produces the element which is the result of letting the algebra element ae act on the representation element re.
‣ QuiverRepresentation( A, dimensions, matrices ) | ( operation ) |
‣ QuiverRepresentationNC( A, dimensions, matrices ) | ( operation ) |
Returns: IsQuiverRepresentation (3.1-2)
Construct a quiver representation over the quiver algebra A. The list dimensions gives the dimension of the vector space in each vertex. The list matrices gives the matrices for the linear maps corresponding to the arrows.
The variant QuiverRepresentationNC does not check that the contents of the lists dimensions and matrices are compatible with each other and with the algebra.
‣ QuiverRepresentationByArrows( A, dimensions, arrows, matrices ) | ( operation ) |
Returns: IsQuiverRepresentation (3.1-2)
Construct a quiver representation by specifying matrices for certain arrows. This works like the constructor QuiverRepresentation (3.5-1), except that the entries in the list matrices correspond to the arrows in the list arrows. All arrows of the quiver that are not present in this list get zero maps in the representation.
‣ AsRepresentationOfQuotientAlgebra( R, A ) | ( operation ) |
Returns: IsQuiverRepresentation (3.1-2)
Given an algebra A = kQ/I and a representation R over the path algebra kQ which respects the relations I, this operation produces the corresponding representation over the algebra A. If the representation R does not respect the relations, then an error is signalled.
‣ ZeroRepresentation( A ) | ( attribute ) |
Returns: IsQuiverRepresentation (3.1-2)
Produces the zero representation over the quiver algebra A.
‣ AlgebraOfRepresentation( R ) | ( attribute ) |
Returns: IsQuiverAlgebra (2.1-2)
Returns the algebra that the representation R is a representation over.
‣ QuiverOfRepresentation( R ) | ( attribute ) |
Returns: IsQuiver (1.1-7)
Returns the quiver that the representation R is a representation over. Calling QuiverOfRepresentation( R ) is equivalent to calling QuiverOfAlgebra( AlgebraOfRepresentation( R ) ).
‣ FieldOfRepresentation( R ) | ( attribute ) |
Returns: IsField (Reference: IsField)
Returns the field that the representation R is a vector space over. This is the same as LeftActingDomain( AlgebraOfRepresentation( R ) ).
‣ VertexDimensions( R ) | ( attribute ) |
Returns: IsDenseList (Reference: IsDenseList)
Returns the dimensions of the vector spaces in the representation R, as a list. The entries in the list correspond to the vertices of the quiver.
‣ VertexDimension( R, i ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the dimension of the vector space in the representation R at vertex number i, where i is a positive integer.
‣ VertexDimension( R, v ) | ( operation ) |
Returns: IsVector (Reference: IsVector)
Returns the dimension of the vector space in the representation R at the vertex v.
‣ MatricesOfRepresentation( R ) | ( attribute ) |
Returns: IsDenseList (Reference: IsDenseList)
Returns a list of the matrices for the maps in the representation R. The list is ordered in the same way as the arrows of the quiver.
‣ MatrixForArrow( R, i ) | ( operation ) |
Returns: IsMatrix (Reference: IsMatrix)
Returns the matrix for the map in the representation R corresponding to arrow number i, where i is a positive integer.
‣ MatrixForArrow( R, a ) | ( operation ) |
Returns: IsMatrix (Reference: IsMatrix)
Returns the matrix for the map in the representation R corresponding to the arrow a.
‣ MatrixForPath( R, p ) | ( operation ) |
Returns: IsMatrix (Reference: IsMatrix)
Returns the matrix for the map in the representation R corresponding to the path p, that is, the composition of the maps corresponding to the arrows in p.
‣ MatrixForAlgebraElement( R, e ) | ( operation ) |
Returns: IsMatrix (Reference: IsMatrix)
Returns the matrix for the map in the representation R corresponding to the homogeneous algebra element e.
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