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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

Certification a gold star

Version 0.1 of this package was accepted for publication in volume 15 of Journal of Software for Algebra and Geometry on 2025-08-20, in the article Quivers and moduli of their thin sincere representations in Macaulay2 (DOI: 10.2140/jsag.2025.15.93). That version can be obtained from the journal.

Version

This documentation describes version 0.1 of ThinSincereQuivers, released January 18, 2025.

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/stable/M2/Macaulay2/packages}}
}

@article{ThinSincereQuiversArticle,
  title = {{Quivers and moduli of their thin sincere representations in \emph{Macaulay2}}},
  author = {Mary Barker and Patricio Gallardo},
  journal = {Journal of Software for Algebra and Geometry},
  volume = {15},
  year = {2025},
}

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 /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/ThinSincereQuivers.m2:3291:0.