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ToricQuiver -- the ToricQuiver datatype

Description

A toric quiver is an acyclic directed graph with edge set $Q_1$ and vertices $Q_0$, equipped with a flow, which is a vector of values associated to each edge of the graph, and a weight, which is a vector of values associated to each vertex in the graph. The weights are obtained from the flow and graph as the image of the incidence map, defined as follows: $$ \text{inc}(\mathbf{w})(i) := \sum_{a\in Q_1 \atop a^{+} = i} \mathbf{w}(a) - \sum_{a\in Q_1 \atop a^{-} = i} \mathbf{w}(a) \quad \text{ for all } i \in Q_0. $$ The ToricQuiver data type is a type of Hash Table with the following keys:

  • IncidenceMatrix:matrix representation of the connected graph underlying the quiver
  • flow: list of integers representing the flow associated to each edge of the quiver
  • Q0: the list of vertices
  • Q1: the list of edges
  • weights: the values on each vertex induced by the flow

See also

Functions and methods returning a toric quiver:

  • mergeOnArrow(Matrix,ZZ,Matrix,ZZ) -- see mergeOnArrow -- join two quivers together by identifying an arrow from each
  • mergeOnArrow(Matrix,ZZ,ToricQuiver,ZZ) -- see mergeOnArrow -- join two quivers together by identifying an arrow from each
  • mergeOnArrow(ToricQuiver,ZZ,Matrix,ZZ) -- see mergeOnArrow -- join two quivers together by identifying an arrow from each
  • mergeOnArrow(ToricQuiver,ZZ,ToricQuiver,ZZ) -- see mergeOnArrow -- join two quivers together by identifying an arrow from each
  • mergeOnVertex(Matrix,ZZ,Matrix,ZZ) -- see mergeOnVertex -- join two quivers together by identifying a vertex from each
  • mergeOnVertex(Matrix,ZZ,ToricQuiver,ZZ) -- see mergeOnVertex -- join two quivers together by identifying a vertex from each
  • mergeOnVertex(ToricQuiver,ZZ,Matrix,ZZ) -- see mergeOnVertex -- join two quivers together by identifying a vertex from each
  • mergeOnVertex(ToricQuiver,ZZ,ToricQuiver,ZZ) -- see mergeOnVertex -- join two quivers together by identifying a vertex from each
  • threeVertexQuiver(List) -- see threeVertexQuiver -- make a toric quiver on underlying graph with three vertices and a specified number of edges between each
  • toricQuiver(List) -- see toricQuiver -- the toricQuiver constructor
  • toricQuiver(List,List) -- see toricQuiver -- the toricQuiver constructor
  • toricQuiver(Matrix) -- see toricQuiver -- the toricQuiver constructor
  • toricQuiver(Matrix,List) -- see toricQuiver -- the toricQuiver constructor
  • toricQuiver(ToricQuiver) -- see toricQuiver -- the toricQuiver constructor
  • toricQuiver(ToricQuiver,List) -- see toricQuiver -- the toricQuiver constructor
  • ToricQuiver ^ List -- taking a subquiver by indexing
  • ToricQuiver _ List -- taking a subquiver by indexing

Methods that use a toric quiver:

  • basisForFlowPolytope(List,ToricQuiver) -- see basisForFlowPolytope -- compute the necessary basis vectors for the hyperplane of a flow polytope
  • basisForFlowPolytope(ToricQuiver) -- see basisForFlowPolytope -- compute the necessary basis vectors for the hyperplane of a flow polytope
  • flowPolytopeVertices(List,ToricQuiver) -- see flowPolytopeVertices -- generate the polytope associated to a toric quiver
  • flowPolytopeVertices(ToricQuiver) -- see flowPolytopeVertices -- generate the polytope associated to a toric quiver
  • getWeights(ToricQuiver) -- see getWeights -- image of the flow on the vertices
  • incInverse(List,ToricQuiver) -- see incInverse -- compute a flow in the preimage for a given weight
  • isAcyclic(ToricQuiver) -- see isAcyclic -- whether a toric quiver has no cycles
  • isClosedUnderArrows(List,ToricQuiver) -- see isClosedUnderArrows -- whether a subquiver closed under arrows
  • isClosedUnderArrows(Matrix,ToricQuiver) -- see isClosedUnderArrows -- whether a subquiver closed under arrows
  • isClosedUnderArrows(ToricQuiver,ToricQuiver) -- see isClosedUnderArrows -- whether a subquiver closed under arrows
  • isSemistable(List,ToricQuiver) -- see isSemistable -- whether a subquiver is semistable with respect to a given weight
  • isSemistable(ToricQuiver,ToricQuiver) -- see isSemistable -- whether a subquiver is semistable with respect to a given weight
  • isStable(List,ToricQuiver) -- see isStable -- whether a subquiver is semistable with respect to a given weight
  • isStable(ToricQuiver,ToricQuiver) -- see isStable -- whether a subquiver is semistable with respect to a given weight
  • isTight(List,ToricQuiver) -- see isTight -- determine if toric quiver is tight
  • isTight(ToricQuiver) -- see isTight -- determine if toric quiver is tight
  • isTight(ToricQuiver,List) -- see isTight -- determine if toric quiver is tight
  • isWellDefined(ToricQuiver) -- determine if toric quiver is correctly constructed
  • maxCodimensionUnstable(ToricQuiver) -- see maxCodimensionUnstable -- compute the maximal codimension of the unstable loci of a quiver
  • potentialWalls(ToricQuiver) -- see potentialWalls -- lists the potential walls in the weight chamber decomposition for a given quiver
  • quiverEdges(ToricQuiver) -- see quiverEdges -- return the graph edges associated to the toric quiver
  • quiverFlow(ToricQuiver) -- see quiverFlow -- return the flow attribute associated to the toric quiver
  • quiverIncidenceMatrix(ToricQuiver) -- see quiverIncidenceMatrix -- return the graph incidence matrix attribute associated to the toric quiver
  • quiverVertices(ToricQuiver) -- see quiverVertices -- return the vertices of the toric quiver
  • quiverWeights(ToricQuiver) -- see quiverWeights -- return the weight attribute associated to the toric quiver
  • samePolytope(List,List,ToricQuiver) -- see samePolytope -- whether two weights produce the same flow polytope
  • subquivers(ToricQuiver) -- see subquivers -- return all possible subquivers of a given quiver
  • ToricQuiver == ToricQuiver -- whether two toric quivers are equal.
  • wallType(List,ToricQuiver) -- see wallType -- get the type of a wall for a given quiver

For the programmer

The object ToricQuiver is a type, with ancestor classes HashTable < Thing.

The source of this document is in ThinSincereQuivers.m2:3279:0.