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ThinSincereQuivers :: ThinSincereQuivers

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 0.0 of ThinSincereQuivers.

Source code

The source code from which this documentation is derived is in the file ThinSincereQuivers.m2.

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
    • flowPolytope -- generate the polytope associated to a toric quiver
    • incInverse -- compute a flow in the preimage for a given weight
    • isAcyclic -- check that a quiver has no cycles
    • isClosedUnderArrows -- is a subquiver closed under arrows?
    • isSemistable -- determines if a subquiver is semistable with respect to a given weight
    • isStable -- determines if 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 -- return 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
    • quiverWeights -- return the weight attribute associated to the toric quiver
    • referenceThetas -- return a weight for all polytopes associated to a toric quiver
    • sameChamber -- determine if two weights lie in the same chamber
    • stableTrees -- return the spanning trees that are stable
    • subquivers -- return all possible subquivers of a given quiver
    • theta -- image of the flow on the vertices
    • threeVertexQuiver -- make a toric quiver on underlying graph with three vertices and a specified number of edges between each
    • toricQuiver -- the toricQuiver constructor
    • 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
    • "flowPolytope(List,ToricQuiver)" -- see flowPolytope -- generate the polytope associated to a toric quiver
    • "flowPolytope(ToricQuiver)" -- see flowPolytope -- generate the polytope associated to a toric quiver
    • "isAcyclic(ToricQuiver)" -- see isAcyclic -- check that a quiver has no cycles
    • "isClosedUnderArrows(List,ToricQuiver)" -- see isClosedUnderArrows -- is a subquiver closed under arrows?
    • "isClosedUnderArrows(Matrix,ToricQuiver)" -- see isClosedUnderArrows -- is a subquiver closed under arrows?
    • "isClosedUnderArrows(ToricQuiver,ToricQuiver)" -- see isClosedUnderArrows -- is a subquiver closed under arrows?
    • "isSemistable(List,ToricQuiver)" -- see isSemistable -- determines if a subquiver is semistable with respect to a given weight
    • "isSemistable(ToricQuiver,ToricQuiver)" -- see isSemistable -- determines if a subquiver is semistable with respect to a given weight
    • "isStable(List,ToricQuiver)" -- see isStable -- determines if a subquiver is semistable with respect to a given weight
    • "isStable(ToricQuiver,ToricQuiver)" -- see isStable -- determines if 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
    • "maxCodimensionUnstable(ToricQuiver)" -- see maxCodimensionUnstable -- compute the maximal codimension of the unstable loci of a quiver
    • "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,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(ToricQuiver)" -- see potentialWalls -- return the potential walls in the weight chamber decomposition for a given quiver
    • "subquivers(ToricQuiver)" -- see subquivers -- return all possible subquivers of a given quiver
    • "theta(ToricQuiver)" -- see theta -- image of the flow on the vertices
    • "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 -- comparing instances of ToricQuiver
    • ToricQuiver ^ List -- taking a subquiver by indexing
    • ToricQuiver _ List -- taking a subquiver by indexing
    • "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.