# Quiver¶

A quiver is an oriented graph without loops, two-cycles, or multiple edges. The edges are labelled by pairs $$(i,-j)$$ (with $$i$$ and $$j$$ being positive integers) such that the matrix $$M = (m_{ab})$$ with $$m_{ab} = i, m_{ba} = -j$$ for an edge $$(i,-j)$$ between vertices $$a$$ and $$b$$ is skew-symmetrizable.

Warning

This is not the standard definition of a quiver. Normally, in cluster algebra theory, a quiver is defined as an oriented graph without loops and two-cycles but with multiple edges allowed; the edges are unlabelled. This notion of quivers, however, can be seen as a particular case of our notion of quivers. Namely, if we have a quiver (in the regular sense of this word) with (precisely) $$i$$ edges from $$a$$ to $$b$$, then we represent it by a quiver (in our sense of this word) with an edge from $$a$$ to $$b$$ labelled by the pair $$(i,-i)$$.

For the compendium on the cluster algebra and quiver package see [MS2011]

AUTHORS:

• Gregg Musiker

• Christian Stump

For mutation types of combinatorial quivers, see QuiverMutationType(). Cluster seeds are closely related to ClusterSeed().

class sage.combinat.cluster_algebra_quiver.quiver.ClusterQuiver(data, frozen=None, user_labels=None)

The quiver associated to an exchange matrix.

INPUT:

• data – can be any of the following:

* QuiverMutationType
* str - a string representing a QuiverMutationType or a common quiver type (see Examples)
* ClusterQuiver
* Matrix - a skew-symmetrizable matrix
* DiGraph - must be the input data for a quiver
* List of edges - must be the edge list of a digraph for a quiver

• frozen – (default:None) sets the list of frozen variables if the input type is a DiGraph, it is ignored otherwise

• user_labels – (default:None) sets the names of the labels for the vertices of the quiver if the input type is not a DiGraph; otherwise it is ignored

EXAMPLES:

From a QuiverMutationType:

sage: Q = ClusterQuiver(['A',5]); Q
Quiver on 5 vertices of type ['A', 5]

sage: Q = ClusterQuiver(['B',2]); Q
Quiver on 2 vertices of type ['B', 2]
sage: Q2 = ClusterQuiver(['C',2]); Q2
Quiver on 2 vertices of type ['B', 2]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
True
sage: MT = Q2.mutation_type(); MT.standard_quiver() == Q2
False

sage: Q = ClusterQuiver(['A',[2,5],1]); Q
Quiver on 7 vertices of type ['A', [2, 5], 1]

sage: Q = ClusterQuiver(['A', [5,0],1]); Q
Quiver on 5 vertices of type ['D', 5]
sage: Q.is_finite()
True
sage: Q.is_acyclic()
False

sage: Q = ClusterQuiver(['F', 4, [2,1]]); Q
Quiver on 6 vertices of type ['F', 4, [1, 2]]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
False
sage: dg = Q.digraph(); Q.mutate([2,1,4,0,5,3])
sage: dg2 = Q.digraph(); dg2.is_isomorphic(dg,edge_labels=True)
False
sage: dg2.is_isomorphic(MT.standard_quiver().digraph(),edge_labels=True)
True

sage: Q = ClusterQuiver(['G',2, (3,1)]); Q
Quiver on 4 vertices of type ['G', 2, [1, 3]]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
False

sage: Q = ClusterQuiver(['GR',[3,6]]); Q
Quiver on 4 vertices of type ['D', 4]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
False

sage: Q = ClusterQuiver(['GR',[3,7]]); Q
Quiver on 6 vertices of type ['E', 6]

sage: Q = ClusterQuiver(['TR',2]); Q
Quiver on 3 vertices of type ['A', 3]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
False
sage: Q.mutate([1,0]); MT.standard_quiver() == Q
True

sage: Q = ClusterQuiver(['TR',3]); Q
Quiver on 6 vertices of type ['D', 6]
sage: MT = Q.mutation_type(); MT.standard_quiver() == Q
False


From a ClusterQuiver:

sage: Q = ClusterQuiver(['A',[2,5],1]); Q
Quiver on 7 vertices of type ['A', [2, 5], 1]
sage: T = ClusterQuiver( Q ); T
Quiver on 7 vertices of type ['A', [2, 5], 1]


From a Matrix:

sage: Q = ClusterQuiver(['A',[2,5],1]); Q
Quiver on 7 vertices of type ['A', [2, 5], 1]
sage: T = ClusterQuiver( Q._M ); T
Quiver on 7 vertices

sage: Q = ClusterQuiver( matrix([[0,1,-1],[-1,0,1],[1,-1,0],[1,2,3]]) ); Q
Quiver on 4 vertices with 1 frozen vertex

sage: Q = ClusterQuiver( matrix([]) ); Q
Quiver without vertices


From a DiGraph:

sage: Q = ClusterQuiver(['A',[2,5],1]); Q
Quiver on 7 vertices of type ['A', [2, 5], 1]
sage: T = ClusterQuiver( Q._digraph ); T
Quiver on 7 vertices

sage: Q = ClusterQuiver( DiGraph([[1,2],[2,3],[3,4],[4,1]]) ); Q
Quiver on 4 vertices

sage: Q = ClusterQuiver(DiGraph([['a', 'b'], ['b', 'c'], ['c', 'd'], ['d', 'e']]),
....:          frozen=['c']); Q
Quiver on 5 vertices with 1 frozen vertex
sage: Q.mutation_type()
[ ['A', 2], ['A', 2] ]
sage: Q
Quiver on 5 vertices of type [ ['A', 2], ['A', 2] ] with 1 frozen vertex


From a List of edges:

sage: Q = ClusterQuiver(['A',[2,5],1]); Q
Quiver on 7 vertices of type ['A', [2, 5], 1]
sage: T = ClusterQuiver( Q._digraph.edges() ); T
Quiver on 7 vertices

sage: Q = ClusterQuiver([[1, 2], [2, 3], [3, 4], [4, 1]]); Q
Quiver on 4 vertices

b_matrix()

Return the b-matrix of self.

EXAMPLES:

sage: ClusterQuiver(['A',4]).b_matrix()
[ 0  1  0  0]
[-1  0 -1  0]
[ 0  1  0  1]
[ 0  0 -1  0]

sage: ClusterQuiver(['B',4]).b_matrix()
[ 0  1  0  0]
[-1  0 -1  0]
[ 0  1  0  1]
[ 0  0 -2  0]

sage: ClusterQuiver(['D',4]).b_matrix()
[ 0  1  0  0]
[-1  0 -1 -1]
[ 0  1  0  0]
[ 0  1  0  0]

sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).b_matrix()
[ 0  1  0  0]
[-1  0  0  0]
[ 0  0  0  1]
[ 0  0 -2  0]

canonical_label(certificate=False)

Return the canonical labelling of self.

INPUT:

• certificate – boolean (default: False) if True, the dictionary from self.vertices() to the vertices of the returned quiver is returned as well.

EXAMPLES:

sage: Q = ClusterQuiver(['A',4]); Q.digraph().edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

sage: T = Q.canonical_label(); T.digraph().edges()
[(0, 3, (1, -1)), (1, 2, (1, -1)), (1, 3, (1, -1))]

sage: T, iso = Q.canonical_label(certificate=True)
sage: T.digraph().edges(); iso
[(0, 3, (1, -1)), (1, 2, (1, -1)), (1, 3, (1, -1))]
{0: 0, 1: 3, 2: 1, 3: 2}

sage: Q = ClusterQuiver(QuiverMutationType([['B',2],['A',1]])); Q
Quiver on 3 vertices of type [ ['B', 2], ['A', 1] ]

sage: Q.canonical_label()
Quiver on 3 vertices of type [ ['A', 1], ['B', 2] ]

sage: Q.canonical_label(certificate=True)
(Quiver on 3 vertices of type [ ['A', 1], ['B', 2] ], {0: 1, 1: 2, 2: 0})

d_vector_fan()

Return the d-vector fan associated with the quiver.

It is the fan whose maximal cones are generated by the d-matrices of the clusters.

This is a complete simplicial fan (and even smooth when the initial quiver is acyclic). It only makes sense for quivers of finite type.

EXAMPLES:

sage: Fd = ClusterQuiver([[1,2]]).d_vector_fan(); Fd
Rational polyhedral fan in 2-d lattice N
sage: Fd.ngenerating_cones()
5

sage: Fd = ClusterQuiver([[1,2],[2,3]]).d_vector_fan(); Fd
Rational polyhedral fan in 3-d lattice N
sage: Fd.ngenerating_cones()
14
sage: Fd.is_smooth()
True

sage: Fd = ClusterQuiver([[1,2],[2,3],[3,1]]).d_vector_fan(); Fd
Rational polyhedral fan in 3-d lattice N
sage: Fd.ngenerating_cones()
14
sage: Fd.is_smooth()
False

digraph()

Return the underlying digraph of self.

EXAMPLES:

sage: ClusterQuiver(['A',1]).digraph()
Digraph on 1 vertex
sage: list(ClusterQuiver(['A',1]).digraph())
[0]
sage: ClusterQuiver(['A',1]).digraph().edges()
[]

sage: ClusterQuiver(['A',4]).digraph()
Digraph on 4 vertices
sage: ClusterQuiver(['A',4]).digraph().edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

sage: ClusterQuiver(['B',4]).digraph()
Digraph on 4 vertices
sage: ClusterQuiver(['A',4]).digraph().edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).digraph()
Digraph on 4 vertices

sage: ClusterQuiver(QuiverMutationType([['A',2],['B',2]])).digraph().edges()
[(0, 1, (1, -1)), (2, 3, (1, -2))]

sage: ClusterQuiver(['C', 4], user_labels = ['x', 'y', 'z', 'w']).digraph().edges()
[('x', 'y', (1, -1)), ('z', 'w', (2, -1)), ('z', 'y', (1, -1))]

exchangeable_part()

Return the restriction to the principal part (i.e. exchangeable part) of self, the subquiver obtained by deleting the frozen vertices of self.

EXAMPLES:

sage: Q = ClusterQuiver(['A',4])
sage: T = ClusterQuiver(Q.digraph().edges(), frozen=[3])
sage: T.digraph().edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]

sage: T.exchangeable_part().digraph().edges()
[(0, 1, (1, -1)), (2, 1, (1, -1))]

sage: Q2 = Q.principal_extension()
sage: Q3 = Q2.principal_extension()
sage: Q2.exchangeable_part() == Q3.exchangeable_part()
True

first_sink()

Return the first vertex of self that is a sink.

EXAMPLES:

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([1,2,4,3,2])
sage: Q.first_sink()
0

first_source()

Return the first vertex of self that is a source

EXAMPLES:

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([2,1,3,4,2])
sage: Q.first_source()
1

free_vertices()

Return the list of free vertices of self.

EXAMPLES:

sage: Q = ClusterQuiver(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]),
....:                   frozen=['b', 'd'])
sage: Q.free_vertices()
['a', 'c', 'e']

frozen_vertices()

Return the list of frozen vertices of self.

EXAMPLES:

sage: Q = ClusterQuiver(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]),
....:                   frozen=['b', 'd'])
sage: sorted(Q.frozen_vertices())
['b', 'd']

g_vector_fan()

Return the g-vector fan associated with the quiver.

It is the fan whose maximal cones are generated by the g-matrices of the clusters.

This is a complete simplicial fan. It is only supported for quivers of finite type.

EXAMPLES:

sage: Fg = ClusterQuiver([[1,2]]).g_vector_fan(); Fg
Rational polyhedral fan in 2-d lattice N
sage: Fg.ngenerating_cones()
5

sage: Fg = ClusterQuiver([[1,2],[2,3]]).g_vector_fan(); Fg
Rational polyhedral fan in 3-d lattice N
sage: Fg.ngenerating_cones()
14
sage: Fg.is_smooth()
True

sage: Fg = ClusterQuiver([[1,2],[2,3],[3,1]]).g_vector_fan(); Fg
Rational polyhedral fan in 3-d lattice N
sage: Fg.ngenerating_cones()
14
sage: Fg.is_smooth()
True

interact(fig_size=1, circular=True)

Start an interactive window for cluster quiver mutations.

Only in Jupyter notebook mode.

INPUT:

• fig_size – (default: 1) factor by which the size of the plot is multiplied.

• circular – (default: True) if True, the circular plot is chosen, otherwise >>spring<< is used.

is_acyclic()

Return true if self is acyclic.

EXAMPLES:

sage: ClusterQuiver(['A',4]).is_acyclic()
True

sage: ClusterQuiver(['A',[2,1],1]).is_acyclic()
True

sage: ClusterQuiver([[0,1],[1,2],[2,0]]).is_acyclic()
False

is_bipartite(return_bipartition=False)

Return True if self is bipartite.

EXAMPLES:

sage: ClusterQuiver(['A',[3,3],1]).is_bipartite()
True

sage: ClusterQuiver(['A',[4,3],1]).is_bipartite()
False

is_finite()

Return True if self is of finite type.

EXAMPLES:

sage: Q = ClusterQuiver(['A',3])
sage: Q.is_finite()
True
sage: Q = ClusterQuiver(['A',[2,2],1])
sage: Q.is_finite()
False
sage: Q = ClusterQuiver([['A',3],['B',3]])
sage: Q.is_finite()
True
sage: Q = ClusterQuiver(['T',[4,4,4]])
sage: Q.is_finite()
False
sage: Q = ClusterQuiver([['A',3],['T',[4,4,4]]])
sage: Q.is_finite()
False
sage: Q = ClusterQuiver([['A',3],['T',[2,2,3]]])
sage: Q.is_finite()
True
sage: Q = ClusterQuiver([['A',3],['D',5]])
sage: Q.is_finite()
True
sage: Q = ClusterQuiver([['A',3],['D',5,1]])
sage: Q.is_finite()
False

sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2]])
sage: Q.is_finite()
False

sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2],[3,4,1],[4,5,1]])
sage: Q.is_finite()
False

is_mutation_finite(nr_of_checks=None, return_path=False)

Uses a non-deterministic method by random mutations in various directions. Can result in a wrong answer.

INPUT:

• nr_of_checks – (default: None) number of mutations applied. Standard is 500*(number of vertices of self).

• return_path – (default: False) if True, in case of self not being mutation finite, a path from self to a quiver with an edge label (a,-b) and a*b > 4 is returned.

ALGORITHM:

A quiver is mutation infinite if and only if every edge label (a,-b) satisfy a*b > 4. Thus, we apply random mutations in random directions

EXAMPLES:

sage: Q = ClusterQuiver(['A',10])
sage: Q._mutation_type = None
sage: Q.is_mutation_finite()
True

sage: Q = ClusterQuiver([(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(2,9)])
sage: Q.is_mutation_finite()
False

m()

Return the number of frozen vertices of self.

EXAMPLES:

sage: Q = ClusterQuiver(['A',4])
sage: Q.m()
0

sage: T = ClusterQuiver(Q.digraph().edges(), frozen=[3])
sage: T.n()
3
sage: T.m()
1

mutate(data, inplace=True)

Mutates self at a sequence of vertices.

INPUT:

• sequence – a vertex of self, an iterator of vertices of self, a function which takes in the ClusterQuiver and returns a vertex or an iterator of vertices, or a string of the parameter wanting to be called on ClusterQuiver that will return a vertex or an iterator of vertices.

• inplace – (default: True) if False, the result is returned, otherwise self is modified.

EXAMPLES:

sage: Q = ClusterQuiver(['A',4]); Q.b_matrix()
[ 0  1  0  0]
[-1  0 -1  0]
[ 0  1  0  1]
[ 0  0 -1  0]

sage: Q.mutate(0); Q.b_matrix()
[ 0 -1  0  0]
[ 1  0 -1  0]
[ 0  1  0  1]
[ 0  0 -1  0]

sage: T = Q.mutate(0, inplace=False); T
Quiver on 4 vertices of type ['A', 4]

sage: Q.mutate(0)
sage: Q == T
True

sage: Q.mutate([0,1,0])
sage: Q.b_matrix()
[ 0 -1  1  0]
[ 1  0  0  0]
[-1  0  0  1]
[ 0  0 -1  0]

sage: Q = ClusterQuiver(QuiverMutationType([['A',1],['A',3]]))
sage: Q.b_matrix()
[ 0  0  0  0]
[ 0  0  1  0]
[ 0 -1  0 -1]
[ 0  0  1  0]

sage: T = Q.mutate(0,inplace=False)
sage: Q == T
True

sage: Q = ClusterQuiver(['A',3]); Q.b_matrix()
[ 0  1  0]
[-1  0 -1]
[ 0  1  0]
sage: Q.mutate('first_sink'); Q.b_matrix()
[ 0 -1  0]
[ 1  0  1]
[ 0 -1  0]
sage: Q.mutate('first_source'); Q.b_matrix()
[ 0  1  0]
[-1  0 -1]
[ 0  1  0]

sage: dg = DiGraph()
sage: Q2 = ClusterQuiver(dg, frozen=['c']); Q2.b_matrix()
[ 0  1  0  0]
[-1  0  0  0]
[ 0  0  0  1]
[ 0  0 -1  0]
[ 0 -1  1  0]
sage: Q2.mutate('a'); Q2.b_matrix()
[ 0 -1  0  0]
[ 1  0  0  0]
[ 0  0  0  1]
[ 0  0 -1  0]
[ 0 -1  1  0]

sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges')
sage: Q = ClusterQuiver(dg);Q
Quiver on 3 vertices
sage: Q.mutate(['a','b'],inplace=False).digraph().edges()
[('a', 'b', (1, -1)), ('c', 'b', (1, -1))]

mutation_class(depth=+ Infinity, show_depth=False, return_paths=False, data_type='quiver', up_to_equivalence=True, sink_source=False)

Return the mutation class of self together with certain constraints.

INPUT:

• depth – (default: infinity) integer, only seeds with distance at most depth from self are returned

• show_depth – (default: False) if True, the actual depth of the mutation is shown

• return_paths – (default: False) if True, a shortest path of mutation sequences from self to the given quiver is returned as well

• data_type – (default: "quiver") can be one of the following:

• "quiver" – the quiver is returned

• "dig6" – the dig6-data is returned

• "path" – shortest paths of mutation sequences from self are returned

• sink_source – (default: False) if True, only mutations at sinks and sources are applied

EXAMPLES:

sage: Q = ClusterQuiver(['A',3])
sage: Ts = Q.mutation_class()
sage: for T in Ts: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: Ts = Q.mutation_class(depth=1)
sage: for T in Ts: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: Ts = Q.mutation_class(show_depth=True)
Depth: 0     found: 1          Time: ... s
Depth: 1     found: 3          Time: ... s
Depth: 2     found: 4          Time: ... s

sage: Ts = Q.mutation_class(return_paths=True)
sage: for T in Ts: print(T)
(Quiver on 3 vertices of type ['A', 3], [])
(Quiver on 3 vertices of type ['A', 3], [1])
(Quiver on 3 vertices of type ['A', 3], [0])
(Quiver on 3 vertices of type ['A', 3], [0, 1])

sage: Ts = Q.mutation_class(up_to_equivalence=False)
sage: for T in Ts: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: Ts = Q.mutation_class(return_paths=True,up_to_equivalence=False)
sage: len(Ts)
14
sage: Ts[0]
(Quiver on 3 vertices of type ['A', 3], [])

sage: Ts = Q.mutation_class(show_depth=True)
Depth: 0     found: 1          Time: ... s
Depth: 1     found: 3          Time: ... s
Depth: 2     found: 4          Time: ... s

sage: Ts = Q.mutation_class(show_depth=True, up_to_equivalence=False)
Depth: 0     found: 1          Time: ... s
Depth: 1     found: 4          Time: ... s
Depth: 2     found: 6          Time: ... s
Depth: 3     found: 10        Time: ... s
Depth: 4     found: 14        Time: ... s

mutation_class_iter(depth=+ Infinity, show_depth=False, return_paths=False, data_type='quiver', up_to_equivalence=True, sink_source=False)

Return an iterator for the mutation class of self together with certain constraints.

INPUT:

• depth – (default: infinity) integer, only quivers with distance at most depth from self are returned.

• show_depth – (default: False) if True, the actual depth of the mutation is shown.

• return_paths – (default: False) if True, a shortest path of mutation sequences from self to the given quiver is returned as well.

• data_type – (default: “quiver”) can be one of the following:

* "quiver"
* "matrix"
* "digraph"
* "dig6"
* "path"

• up_to_equivalence – (default: True) if True, only one quiver for each graph-isomorphism class is recorded.

• sink_source – (default: False) if True, only mutations at sinks and sources are applied.

EXAMPLES:

sage: Q = ClusterQuiver(['A',3])
sage: it = Q.mutation_class_iter()
sage: for T in it: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: it = Q.mutation_class_iter(depth=1)
sage: for T in it: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: it = Q.mutation_class_iter(show_depth=True)
sage: for T in it: pass
Depth: 0     found: 1          Time: ... s
Depth: 1     found: 3          Time: ... s
Depth: 2     found: 4          Time: ... s

sage: it = Q.mutation_class_iter(return_paths=True)
sage: for T in it: print(T)
(Quiver on 3 vertices of type ['A', 3], [])
(Quiver on 3 vertices of type ['A', 3], [1])
(Quiver on 3 vertices of type ['A', 3], [0])
(Quiver on 3 vertices of type ['A', 3], [0, 1])

sage: it = Q.mutation_class_iter(up_to_equivalence=False)
sage: for T in it: print(T)
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]
Quiver on 3 vertices of type ['A', 3]

sage: it = Q.mutation_class_iter(return_paths=True,up_to_equivalence=False)
sage: mutation_class = list(it)
sage: len(mutation_class)
14
sage: mutation_class[0]
(Quiver on 3 vertices of type ['A', 3], [])

sage: Q = ClusterQuiver(['A',3])
sage: it = Q.mutation_class_iter(data_type='path')
sage: for T in it: print(T)
[]
[1]
[0]
[0, 1]

sage: Q = ClusterQuiver(['A',3])
sage: it = Q.mutation_class_iter(return_paths=True,data_type='matrix')
sage: next(it)
(
[ 0  0  1]
[ 0  0  1]
[-1 -1  0], []
)

sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges')
sage: S = ClusterQuiver(dg, frozen=['b'])
sage: S.mutation_class()
[Quiver on 3 vertices with 1 frozen vertex,
Quiver on 3 vertices with 1 frozen vertex,
Quiver on 3 vertices with 1 frozen vertex]

mutation_sequence(sequence, show_sequence=False, fig_size=1.2)

Return a list containing the sequence of quivers obtained from self by a sequence of mutations on vertices.

INPUT:

• sequence – a list or tuple of vertices of self.

• show_sequence – (default: False) if True, a png containing the mutation sequence is shown.

• fig_size – (default: 1.2) factor by which the size of the sequence is expanded.

EXAMPLES:

sage: Q = ClusterQuiver(['A',4])
sage: seq = Q.mutation_sequence([0,1]); seq
[Quiver on 4 vertices of type ['A', 4], Quiver on 4 vertices of type ['A', 4], Quiver on 4 vertices of type ['A', 4]]
sage: [T.b_matrix() for T in seq]
[
[ 0  1  0  0]  [ 0 -1  0  0]  [ 0  1 -1  0]
[-1  0 -1  0]  [ 1  0 -1  0]  [-1  0  1  0]
[ 0  1  0  1]  [ 0  1  0  1]  [ 1 -1  0  1]
[ 0  0 -1  0], [ 0  0 -1  0], [ 0  0 -1  0]
]

mutation_type()

Return the mutation type of self.

Return the mutation_type of each connected component of self if it can be determined, otherwise, the mutation type of this component is set to be unknown.

The mutation types of the components are ordered by vertex labels.

If you do many type recognitions, you should consider to save exceptional mutation types using save_quiver_data()

WARNING:

• All finite types can be detected,

• All affine types can be detected, EXCEPT affine type D (the algorithm is not yet implemented)

• All exceptional types can be detected.

EXAMPLES:

sage: ClusterQuiver(['A',4]).mutation_type()
['A', 4]
sage: ClusterQuiver(['A',(3,1),1]).mutation_type()
['A', [1, 3], 1]
sage: ClusterQuiver(['C',2]).mutation_type()
['B', 2]
sage: ClusterQuiver(['B',4,1]).mutation_type()
['BD', 4, 1]


finite types:

sage: Q = ClusterQuiver(['A',5])
sage: Q._mutation_type = None
sage: Q.mutation_type()
['A', 5]

sage: Q = ClusterQuiver([(0,1),(1,2),(2,3),(3,4)])
sage: Q.mutation_type()
['A', 5]

sage: Q = ClusterQuiver(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]),
....:                   frozen=['c'])
sage: Q.mutation_type()
[ ['A', 2], ['A', 2] ]


affine types:

sage: Q = ClusterQuiver(['E',8,[1,1]]); Q
Quiver on 10 vertices of type ['E', 8, [1, 1]]
sage: Q._mutation_type = None; Q
Quiver on 10 vertices
sage: Q.mutation_type() # long time
['E', 8, [1, 1]]


the not yet working affine type D (unless user has saved small classical quiver data):

sage: Q = ClusterQuiver(['D',4,1])
sage: Q._mutation_type = None
sage: Q.mutation_type() # todo: not implemented
['D', 4, 1]


the exceptional types:

sage: Q = ClusterQuiver(['X',6])
sage: Q._mutation_type = None
sage: Q.mutation_type() # long time
['X', 6]


examples from page 8 of [Ke2008]:

sage: dg = DiGraph(); dg.add_edges([(9,0),(9,4),(4,6),(6,7),(7,8),(8,3),(3,5),(5,6),(8,1),(2,3)])
sage: ClusterQuiver( dg ).mutation_type() # long time
['E', 8, [1, 1]]

sage: dg = DiGraph( { 0:[3], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7] } )
sage: ClusterQuiver( dg ).mutation_type() # long time
['E', 8, 1]

sage: dg = DiGraph( { 0:[3,9], 1:[0,4], 2:[0,6], 3:[1,2,7], 4:[3,8], 5:[2], 6:[3,5], 7:[4,6], 8:[7], 9:[1] } )
sage: ClusterQuiver( dg ).mutation_type() # long time
['E', 8, [1, 1]]


infinite types:

sage: Q = ClusterQuiver(['GR',[4,9]])
sage: Q._mutation_type = None
sage: Q.mutation_type()
'undetermined infinite mutation type'


reducible types:

sage: Q = ClusterQuiver([['A', 3], ['B', 3]])
sage: Q._mutation_type = None
sage: Q.mutation_type()
[ ['A', 3], ['B', 3] ]

sage: Q = ClusterQuiver([['A', 3], ['T', [4,4,4]]])
sage: Q._mutation_type = None
sage: Q.mutation_type()
[['A', 3], 'undetermined infinite mutation type']

sage: Q = ClusterQuiver([['A', 3], ['B', 3], ['T', [4,4,4]]])
sage: Q._mutation_type = None
sage: Q.mutation_type()
[['A', 3], ['B', 3], 'undetermined infinite mutation type']

sage: Q = ClusterQuiver([[0,1,2],[1,2,2],[2,0,2],[3,4,1],[4,5,1]])
sage: Q.mutation_type()
['undetermined finite mutation type', ['A', 3]]

n()

Return the number of free vertices of self.

EXAMPLES:

sage: ClusterQuiver(['A',4]).n()
4
sage: ClusterQuiver(['A',(3,1),1]).n()
4
sage: ClusterQuiver(['B',4]).n()
4
sage: ClusterQuiver(['B',4,1]).n()
5

number_of_edges()

Return the total number of edges on the quiver

Note: This only works with non-valued quivers. If used on a non-valued quiver then the positive value is taken to be the number of edges added

OUTPUT:

An integer of the number of edges.

EXAMPLES:

sage: S = ClusterQuiver(['A',4]); S.number_of_edges()
3

sage: S = ClusterQuiver(['B',4]); S.number_of_edges()
3

plot(circular=True, center=(0, 0), directed=True, mark=None, save_pos=False, greens=[])

Return the plot of the underlying digraph of self.

INPUT:

• circular – (default: True) if True, the circular plot is chosen, otherwise >>spring<< is used.

• center – (default:(0,0)) sets the center of the circular plot, otherwise it is ignored.

• directed – (default: True) if True, the directed version is shown, otherwise the undirected.

• mark – (default: None) if set to i, the vertex i is highlighted.

• save_pos – (default: False) if True, the positions of the vertices are saved.

• greens – (default: []) if set to a list, will display the green vertices as green

EXAMPLES:

sage: Q = ClusterQuiver(['A',5])
sage: Q.plot()
Graphics object consisting of 15 graphics primitives
sage: Q.plot(circular=True)
Graphics object consisting of 15 graphics primitives
sage: Q.plot(circular=True, mark=1)
Graphics object consisting of 15 graphics primitives

poincare_semistable(theta, d)

Return the Poincaré polynomial of the moduli space of semi-stable representations of dimension vector $$d$$.

INPUT:

• theta – stability weight, as list or vector of rationals

• d – dimension vector, as list or vector of coprime integers

The semi-stability is taken with respect to the slope function

$\mu(d) = \theta(d) / \operatorname{dim}(d)$

where $$d$$ is a dimension vector.

This uses the matrix-inversion algorithm from [Rei2002].

EXAMPLES:

sage: Q = ClusterQuiver(['A',2])
sage: Q.poincare_semistable([1,0],[1,0])
1
sage: Q.poincare_semistable([1,0],[1,1])
1

sage: K2 = ClusterQuiver(matrix([[0,2],[-2,0]]))
sage: theta = (1, 0)
sage: K2.poincare_semistable(theta, [1,0])
1
sage: K2.poincare_semistable(theta, [1,1])
v^2 + 1
sage: K2.poincare_semistable(theta, [1,2])
1
sage: K2.poincare_semistable(theta, [1,3])
0

sage: K3 = ClusterQuiver(matrix([[0,3],[-3,0]]))
sage: theta = (1, 0)
sage: K3.poincare_semistable(theta, (2,3))
v^12 + v^10 + 3*v^8 + 3*v^6 + 3*v^4 + v^2 + 1
sage: K3.poincare_semistable(theta, (3,4))(1)
68


REFERENCES:

Rei2002

Markus Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, arXiv math/0204059

principal_extension(inplace=False)

Return the principal extension of self, adding n frozen vertices to any previously frozen vertices. I.e., the quiver obtained by adding an outgoing edge to every mutable vertex of self.

EXAMPLES:

sage: Q = ClusterQuiver(['A',2]); Q
Quiver on 2 vertices of type ['A', 2]
sage: T = Q.principal_extension(); T
Quiver on 4 vertices of type ['A', 2] with 2 frozen vertices
sage: T2 = T.principal_extension(); T2
Quiver on 6 vertices of type ['A', 2] with 4 frozen vertices
sage: Q.digraph().edges()
[(0, 1, (1, -1))]
sage: T.digraph().edges()
[(0, 1, (1, -1)), (2, 0, (1, -1)), (3, 1, (1, -1))]
sage: T2.digraph().edges()
[(0, 1, (1, -1)), (2, 0, (1, -1)), (3, 1, (1, -1)), (4, 0, (1, -1)), (5, 1, (1, -1))]

qmu_save(filename=None)

Save self in a .qmu file.

This file can then be opened in Bernhard Keller’s Quiver Applet.

INPUT:

• filename – the filename the image is saved to.

If a filename is not specified, the default name is from_sage.qmu in the current sage directory.

EXAMPLES:

sage: Q = ClusterQuiver(['F',4,[1,2]])
sage: Q.qmu_save(os.path.join(SAGE_TMP, 'sage.qmu'))


Make sure we can save quivers with $$m != n$$ frozen variables, see trac ticket #14851:

sage: S = ClusterSeed(['A',3])
sage: T1 = S.principal_extension()
sage: Q = T1.quiver()
sage: Q.qmu_save(os.path.join(SAGE_TMP, 'sage.qmu'))

relabel(relabelling, inplace=True)

Return the quiver after doing a relabelling

Will relabel the vertices of the quiver

INPUT:

• relabelling – Dictionary of labels to move around

• inplace – (default:True) if True, will return a duplicate of the quiver

EXAMPLES:

sage: S = ClusterQuiver(['A',4]).relabel({1:'5',2:'go'})

reorient(data)

Reorient self with respect to the given total order, or with respect to an iterator of edges in self to be reverted.

Warning

This operation might change the mutation type of self.

INPUT:

• data – an iterator defining a total order on self.vertices(), or an iterator of edges in self to be reoriented.

EXAMPLES:

sage: Q = ClusterQuiver(['A',(2,3),1])
sage: Q.mutation_type()
['A', [2, 3], 1]

sage: Q.reorient([(0,1),(1,2),(2,3),(3,4)])
sage: Q.mutation_type()
['D', 5]

sage: Q.reorient([0,1,2,3,4])
sage: Q.mutation_type()
['A', [1, 4], 1]

save_image(filename, circular=False)

Save the plot of the underlying digraph of self.

INPUT:

• filename – the filename the image is saved to.

• circular – (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.

EXAMPLES:

sage: Q = ClusterQuiver(['F',4,[1,2]])
sage: Q.save_image(os.path.join(SAGE_TMP, 'sage.png'))

show(fig_size=1, circular=False, directed=True, mark=None, save_pos=False, greens=[])

Show the plot of the underlying digraph of self.

INPUT:

• fig_size – (default: 1) factor by which the size of the plot is multiplied.

• circular – (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.

• directed – (default: True) if True, the directed version is shown, otherwise the undirected.

• mark – (default: None) if set to i, the vertex i is highlighted.

• save_pos – (default:False) if True, the positions of the vertices are saved.

• greens – (default:[]) if set to a list, will display the green vertices as green

sinks()

Return all vertices of self that are sinks.

EXAMPLES:

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([1,2,4,3,2])
sage: Q.sinks()
[0, 2]

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([2,1,3,4,2])
sage: Q.sinks()
[3]

sources()

Return all vertices of self that are sources.

EXAMPLES:

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([1,2,4,3,2])
sage: Q.sources()
[]

sage: Q = ClusterQuiver(['A',5])
sage: Q.mutate([2,1,3,4,2])
sage: Q.sources()
[1]
`