def representation(self, k, *args, **kwds):
"""
Return a representation of the quiver.
For more information see the
:class:`~sage.quivers.representation.QuiverRep` documentation.
TESTS::
sage: Q = DiGraph({1:{3:['a']}, 2:{3:['b']}}).path_semigroup()
sage: spaces = {1: QQ^2, 2: QQ^3, 3: QQ^2}
sage: maps = {(1, 3, 'a'): (QQ^2).Hom(QQ^2).identity(), (2, 3, 'b'): [[1, 0], [0, 0], [0, 0]]}
sage: M = Q.representation(QQ, spaces, maps)
sage: M
Representation with dimension vector (2, 3, 2)
"""
return QuiverRep(k, self, *args, **kwds)
def S(self, k, vertex):
"""
Return the simple module over `k` at the given vertex
``vertex``.
This module is literally simple only when `k` is a field.
INPUT:
- `k` -- ring, the base ring of the representation
- ``vertex`` -- integer, a vertex of the quiver
OUTPUT:
- :class:`~sage.quivers.representation.QuiverRep`, the simple module
at ``vertex`` with base ring `k`
EXAMPLES::
sage: P = DiGraph({1:{2:['a','b']}, 2:{3:['c','d']}}).path_semigroup()
sage: S1 = P.S(GF(3), 1)
sage: P.S(ZZ, 3).dimension_vector()
(0, 0, 1)
sage: P.S(ZZ, 1).dimension_vector()
(1, 0, 0)
The vertex given must be a vertex of the quiver::
sage: P.S(QQ, 4)
Traceback (most recent call last):
...
ValueError: must specify a valid vertex of the quiver
"""
if vertex not in self._quiver:
raise ValueError("must specify a valid vertex of the quiver")
# This is the module with k at the given vertex and zero elsewhere. As
# all maps are zero we only need to specify that the given vertex has
# dimension 1 and the constructor will zero out everything else.
return QuiverRep(k, self, {vertex: 1})
def free_module(self, k):
"""
Return a free module of rank `1` over ``kP``, where `P` is
``self``. (In other words, the regular representation.)
INPUT:
- ``k`` -- ring, the base ring of the representation.
OUTPUT:
- :class:`~sage.quivers.representation.QuiverRep_with_path_basis`, the
path algebra considered as a right module over itself.
EXAMPLES::
sage: Q = DiGraph({1:{2:['a', 'b'], 3: ['c', 'd']}, 2:{3:['e']}}).path_semigroup()
sage: Q.free_module(GF(3)).dimension_vector()
(1, 3, 6)
"""
return QuiverRep(k, self, [[(v, v)] for v in self._quiver], option='paths')
def I(self, k, vertex):
"""
Return the indecomposable injective module over `k` at the
given vertex ``vertex``.
This module is literally indecomposable only when `k` is a field.
INPUT:
- `k` -- ring, the base ring of the representation
- ``vertex`` -- integer, a vertex of the quiver
OUTPUT:
- :class:`~sage.quivers.representation.QuiverRep`, the indecomposable
injective module at vertex ``vertex`` with base ring `k`
EXAMPLES::
sage: Q = DiGraph({1:{2:['a','b']}, 2:{3:['c','d']}}).path_semigroup()
sage: I2 = Q.I(GF(3), 2)
sage: Q.I(ZZ, 3).dimension_vector()
(4, 2, 1)
sage: Q.I(ZZ, 1).dimension_vector()
(1, 0, 0)
The vertex given must be a vertex of the quiver::
sage: Q.I(QQ, 4)
Traceback (most recent call last):
...
ValueError: must specify a valid vertex of the quiver
"""
if vertex not in self._quiver:
raise ValueError("must specify a valid vertex of the quiver")
return QuiverRep(k, self, [[(vertex, vertex)]], option='dual paths')
