def _yang_baxter_graph(self):
r"""
Construct and cache the underlying Yang-Baxter graph.
EXAMPLES::
sage: rho = SymmetricGroupRepresentation([3,2], 'specht')
sage: rho._yang_baxter_graph
Yang-Baxter graph of [3, 2], with top vertex (1, 0, 2, 1, 0)
"""
return YangBaxterGraph_partition(self._partition)
def _yang_baxter_graph(self):
r"""
Return the Yang-Baxter graph associated with the representation,
with vertices labelled by the vector of contents of the partition.
EXAMPLES::
sage: orth = SymmetricGroupRepresentation([3,2], "orthogonal")
sage: orth._yang_baxter_graph
Yang-Baxter graph of [3, 2], with top vertex (0, -1, 2, 1, 0)
"""
Y = YangBaxterGraph_partition(self._partition)
n = self._partition.size()
# relabel vertices with "vector of contents"
Y.relabel_vertices(\
partition_to_vector_of_contents(self._partition, reverse=True))
# relabel edges with "differences"
edge_relabel_dict = {}
for (u, v, op) in Y.edges():
i = op.position() + 1
edge_relabel_dict[u, v] = (n - i, QQ((1, u[i] - u[i - 1])))
Y.relabel_edges(edge_relabel_dict)
return Y
def _yang_baxter_graph(self):
r"""
Return the Yang-Baxter graph associated with the representation,
with vertices labelled by the vector of contents of the partition.
EXAMPLES::
sage: orth = SymmetricGroupRepresentation([3,2], "orthogonal")
sage: orth._yang_baxter_graph
Yang-Baxter graph of [3, 2], with top vertex (0, -1, 2, 1, 0)
"""
Y = YangBaxterGraph_partition(self._partition)
n = self._partition.size()
# relabel vertices with "vector of contents"
Y.relabel_vertices(partition_to_vector_of_contents(self._partition, reverse=True))
# relabel edges with "differences"
edge_relabel_dict = {}
for (u, v, op) in Y.edges():
i = op.position() + 1
edge_relabel_dict[u, v] = (n - i, QQ((1, u[i] - u[i - 1])))
Y.relabel_edges(edge_relabel_dict)
return Y