def _digraph_mutate( dg, k, n, m ):
"""
Returns a digraph obtained from dg with n+m vertices by mutating at vertex k.
INPUT:
- ``dg`` -- a digraph with integral edge labels with ``n+m`` vertices
- ``k`` -- the vertex at which ``dg`` is mutated
EXAMPLES::
sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _digraph_mutate
sage: dg = ClusterQuiver(['A',4]).digraph()
sage: dg.edges()
[(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))]
sage: _digraph_mutate(dg,2,4,0).edges()
[(0, 1, (1, -1)), (1, 2, (1, -1)), (3, 2, (1, -1))]
"""
edges = dict( ((v1,v2),label) for v1,v2,label in dg._backend.iterator_in_edges(dg,True) )
in_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v2 == k ]
out_edges = [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if v1 == k ]
in_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in in_edges ]
out_edges_new = [ (v2,v1,(-label[1],-label[0])) for (v1,v2,label) in out_edges ]
diag_edges_new = []
diag_edges_del = []
for (v1,v2,label1) in in_edges:
for (w1,w2,label2) in out_edges:
l11,l12 = label1
l21,l22 = label2
if (v1,w2) in edges:
diag_edges_del.append( (v1,w2,edges[(v1,w2)]) )
a,b = edges[(v1,w2)]
a,b = a+l11*l21, b-l12*l22
diag_edges_new.append( (v1,w2,(a,b)) )
elif (w2,v1) in edges:
diag_edges_del.append( (w2,v1,edges[(w2,v1)]) )
a,b = edges[(w2,v1)]
a,b = b+l11*l21, a-l12*l22
if a<0:
diag_edges_new.append( (w2,v1,(b,a)) )
elif a>0:
diag_edges_new.append( (v1,w2,(a,b)) )
else:
a,b = l11*l21,-l12*l22
diag_edges_new.append( (v1,w2,(a,b)) )
del_edges = in_edges + out_edges + diag_edges_del
new_edges = in_edges_new + out_edges_new + diag_edges_new
new_edges += [ (v1,v2,edges[(v1,v2)]) for (v1,v2) in edges if not (v1,v2,edges[(v1,v2)]) in del_edges ]
dg_new = DiGraph()
for v1,v2,label in new_edges:
dg_new._backend.add_edge(v1,v2,label,True)
if dg_new.order() < n+m:
dg_new_vertices = [ v for v in dg_new ]
for i in [ v for v in dg if v not in dg_new_vertices ]:
dg_new.add_vertex(i)
return dg_new
def _matrix_to_digraph( M ):
"""
Returns the digraph obtained from the matrix ``M``.
In order to generate a quiver, we assume that ``M`` is skew-symmetrizable.
EXAMPLES::
sage: from sage.combinat.cluster_algebra_quiver.mutation_class import _matrix_to_digraph
sage: _matrix_to_digraph(matrix(3,[0,1,0,-1,0,-1,0,1,0]))
Digraph on 3 vertices
"""
n = M.ncols()
dg = DiGraph(sparse=True)
for i,j in M.nonzero_positions():
if i >= n: a,b = M[i,j],-M[i,j]
else: a,b = M[i,j],M[j,i]
if a > 0:
dg._backend.add_edge(i,j,(a,b),True)
elif i >= n:
dg._backend.add_edge(j,i,(-a,-b),True)
if dg.order() < M.nrows():
for i in [ index for index in xrange(M.nrows()) if index not in dg ]:
dg.add_vertex(i)
return dg
def to_dag(self):
"""
Returns a directed acyclic graph corresponding to the skew
partition.
EXAMPLES::
sage: dag = SkewPartition([[3, 2, 1], [1, 1]]).to_dag()
sage: dag.edges()
[('0,1', '0,2', None), ('0,1', '1,1', None)]
sage: dag.vertices()
['0,1', '0,2', '1,1', '2,0']
"""
i = 0
#Make the skew tableau from the shape
skew = [[1]*row_length for row_length in self.outer()]
inner = self.inner()
for i in range(len(inner)):
for j in range(inner[i]):
skew[i][j] = None
G = DiGraph()
for row in range(len(skew)):
for column in range(len(skew[row])):
if skew[row][column] is not None:
string = "%d,%d" % (row, column)
G.add_vertex(string)
#Check to see if there is a node to the right
if column != len(skew[row]) - 1:
newstring = "%d,%d" % (row, column+1)
G.add_edge(string, newstring)
#Check to see if there is anything below
if row != len(skew) - 1:
if len(skew[row+1]) > column:
if skew[row+1][column] is not None:
newstring = "%d,%d" % (row+1, column)
G.add_edge(string, newstring)
return G
def to_dag(self):
"""
Returns a directed acyclic graph corresponding to the skew
partition.
EXAMPLES::
sage: dag = SkewPartition([[3, 2, 1], [1, 1]]).to_dag()
sage: dag.edges()
[('0,1', '0,2', None), ('0,1', '1,1', None)]
sage: dag.vertices()
['0,1', '0,2', '1,1', '2,0']
"""
i = 0
#Make the skew tableau from the shape
skew = [[1] * row_length for row_length in self.outer()]
inner = self.inner()
for i in range(len(inner)):
for j in range(inner[i]):
skew[i][j] = None
G = DiGraph()
for row in range(len(skew)):
for column in range(len(skew[row])):
if skew[row][column] is not None:
string = "%d,%d" % (row, column)
G.add_vertex(string)
#Check to see if there is a node to the right
if column != len(skew[row]) - 1:
newstring = "%d,%d" % (row, column + 1)
G.add_edge(string, newstring)
#Check to see if there is anything below
if row != len(skew) - 1:
if len(skew[row + 1]) > column:
if skew[row + 1][column] is not None:
newstring = "%d,%d" % (row + 1, column)
G.add_edge(string, newstring)
return G
def _digraph(self):
r"""
Constructs the underlying digraph and stores the result as an
attribute.
EXAMPLES::
sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
sage: ops = [SwapIncreasingOperator(i) for i in range(2)]
sage: Y = YangBaxterGraph(root=(1,2,3), operators=ops)
sage: Y._digraph
Digraph on 6 vertices
"""
digraph = DiGraph()
digraph.add_vertex(self._root)
queue = [self._root]
while queue:
u = queue.pop()
for (v, l) in self._succesors(u):
if v not in digraph:
queue.append(v)
digraph.add_edge(u, v, l)
return digraph
def _digraph(self):
r"""
Constructs the underlying digraph and stores the result as an
attribute.
EXAMPLES::
sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
sage: ops = [SwapIncreasingOperator(i) for i in range(2)]
sage: Y = YangBaxterGraph(root=(1,2,3), operators=ops)
sage: Y._digraph
Digraph on 6 vertices
"""
digraph = DiGraph()
digraph.add_vertex(self._root)
queue = [self._root]
while queue:
u = queue.pop()
for (v, l) in self._succesors(u):
if v not in digraph:
queue.append(v)
digraph.add_edge(u, v, l)
return digraph