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ThinSincereQuivers -- creating and manipulating Toric Quivers

Description

ThinSincereQuivers is a package for creating and manipulating toric quivers.

For further details in the theory, we suggest the following articles and the references within them:

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Authors

Version

This documentation describes version 0.1 of ThinSincereQuivers.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ThinSincereQuiversSource,
  title = {{ThinSincereQuivers: Construction of flow polytopes and their associated quivers. Version~0.1}},
  author = {Mary Barker and Patricio Gallardo},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Types
  • Functions and commands
    • allSpanningTrees -- find the spanning trees of the underlying graph
    • basisForFlowPolytope -- compute the necessary basis vectors for the hyperplane of a flow polytope
    • bipartiteQuiver -- make a toric quiver on underlying bipartite graph
    • chainQuiver -- make a toric quiver on underlying graph in the form of a chain
    • coneSystem -- compute the chamber decomposition of weights
    • flowPolytopeVertices -- generate the polytope associated to a toric quiver
    • getWeights -- image of the flow on the vertices
    • incInverse -- compute a flow in the preimage for a given weight
    • isAcyclic -- whether a toric quiver has no cycles
    • isClosedUnderArrows -- whether a subquiver closed under arrows
    • isSemistable -- whether a subquiver is semistable with respect to a given weight
    • isStable -- whether a subquiver is semistable with respect to a given weight
    • isTight -- determine if toric quiver is tight
    • makeTight -- return a tight quiver with the same flow polytope
    • maxCodimensionUnstable -- compute the maximal codimension of the unstable loci of a quiver
    • maximalNonstableSubquivers -- return the maximal subquivers that are semistable
    • maximalUnstableSubquivers -- return the maximal subquivers that are unstable
    • mergeOnArrow -- join two quivers together by identifying an arrow from each
    • mergeOnVertex -- join two quivers together by identifying a vertex from each
    • potentialWalls -- lists the potential walls in the weight chamber decomposition for a given quiver
    • primitiveArrows -- list the primitive arrows in a quiver
    • quiverEdges -- return the graph edges associated to the toric quiver
    • quiverFlow -- return the flow attribute associated to the toric quiver
    • quiverIncidenceMatrix -- return the graph incidence matrix attribute associated to the toric quiver
    • quiverVertices -- return the vertices of the toric quiver
    • quiverWeights -- return the weight attribute associated to the toric quiver
    • referenceThetas -- return a weight for all polytopes associated to a toric quiver
    • referenceWeights -- return a weight for all polytopes associated to a toric quiver
    • samePolytope -- whether two weights produce the same flow polytope
    • stableTrees -- return the spanning trees that are stable
    • subquivers -- return all possible subquivers of a given quiver
    • threeVertexQuiver -- make a toric quiver on underlying graph with three vertices and a specified number of edges between each
    • toricQuiver -- the toricQuiver constructor
    • wallQPlus -- get the partition of the set of vertices induced by a wall
    • wallType -- get the type of a wall for a given quiver
  • Methods
    • 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(Matrix) -- see getWeights -- image of the flow on the vertices
    • 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(Matrix) -- see isAcyclic -- whether a toric quiver has no cycles
    • isAcyclic(ToricQuiver) -- see isAcyclic -- whether a toric quiver has no cycles
    • isClosedUnderArrows(List,Matrix) -- see isClosedUnderArrows -- whether a subquiver closed under arrows
    • isClosedUnderArrows(List,ToricQuiver) -- see isClosedUnderArrows -- whether a subquiver closed under arrows
    • isClosedUnderArrows(Matrix,List) -- 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
    • 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
    • potentialWalls(Matrix) -- see potentialWalls -- lists the potential walls in the weight chamber decomposition for a given 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
    • referenceThetas(List) -- see referenceThetas -- return a weight for all polytopes associated to a toric quiver
    • referenceWeights(List) -- see referenceWeights -- return a weight for all polytopes associated to a toric quiver
    • samePolytope(List,List,ToricQuiver) -- see samePolytope -- whether two weights produce the same flow polytope
    • subquivers(Matrix) -- see subquivers -- return all possible subquivers of a given quiver
    • subquivers(ToricQuiver) -- see subquivers -- return all possible subquivers of a given quiver
    • 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 == ToricQuiver -- whether two toric quivers are equal.
    • ToricQuiver ^ List -- taking a subquiver by indexing
    • ToricQuiver _ List -- taking a subquiver by indexing
    • wallType(List,Matrix) -- see wallType -- get the type of a wall for a given quiver
    • wallType(List,ToricQuiver) -- see wallType -- get the type of a wall for a given quiver
  • Symbols

For the programmer

The object ThinSincereQuivers is a package, defined in ThinSincereQuivers.m2.

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