Auslander-Reiten Quivers

class sage.quivers.ar_quiver.AuslanderReitenQuiver(quiver)

Bases: UniqueRepresentation, Parent

The Auslander-Reiten quiver.

Let \(Q = (Q_0, Q_1)\) be a finite acyclic quiver. The Auslander-Reiten quiver (AR quiver) \(\Gamma_Q\) is the quiver whose vertices correspond to the indecompositible modules of \(Q\) (equivalently its path algebra over an algebraically closed field) and edges are irreducible morphisms.

In this implementation, we denote the vertices of \(\Gamma_Q\) as certain pairs \(\langle v, k \rangle\), where \(v \in Q_0\) and \(k \in \ZZ \setminus \{0\}\) is called the level. When \(k > 0\) (resp. \(k < 0\)), then it corresponds to a preprojective (resp. postinjective) module. When the quiver is a finite type Dynkin quiver, we consider all modules to be preprojectives and denoted by a positive level.

Note

We use the terminology postinjective instead of preinjective given that they follow from injectives by AR translation.

ALGORITHM:

We compute the dimension vectors of a projective \(\langle v, 1 \rangle\) by counting the number of (directed) paths \(u \to v\) in \(Q\). We then proceed inductively to compute all of the dimension vectors of level \(k\) by using the translation equation

\[dim \langle v, k-1 \rangle + \dim \langle v, k \rangle = \sum_{u,k'} \dim \langle u, k' \rangle,\]

where the sum is over all paths from \(\langle v, k-1 \rangle\) to \(\langle v, k \rangle\) in \(\Gamma_Q\). More specifically, for each edge \((u, v, \ell) \in Q_1\) (resp. \((v, u, \ell) \in Q_1\)), we have \(\langle u, k-1 \rangle\) (resp. \(\langle u, k \rangle\)) in the sum (assuming the node is in the AR quiver).

The algorithm for postinjectives is dual to the above.

Todo

This only is implemented for the preprojectives and postinjectives when the quiver is not a finite type Dynkin quiver.

Todo

Implement this for general Artinian algebras.

EXAMPLES:

We create the AR quivers for finite type \(A_3\) Dynkin quivers:

sage: DA = DiGraph([[1, 2], [2, 3]])
sage: AR = DA.auslander_reiten_quiver()
sage: AR.digraph().edges(labels=False)
[(<1, 1>, <2, 2>), (<2, 1>, <1, 1>), (<2, 1>, <3, 2>), (<3, 1>, <2, 1>),
 (<2, 2>, <3, 3>), (<3, 2>, <2, 2>)]

sage: DA = DiGraph([[1, 2], [3, 2]])
sage: AR = DA.auslander_reiten_quiver()
sage: AR.digraph().edges(labels=False)
[(<1, 1>, <2, 2>), (<2, 1>, <1, 1>), (<2, 1>, <3, 1>), (<3, 1>, <2, 2>),
 (<2, 2>, <1, 2>), (<2, 2>, <3, 2>)]

sage: DA = DiGraph([[2, 1], [2, 3]])
sage: AR = DA.auslander_reiten_quiver()
sage: AR.digraph().edges(labels=False)
[(<1, 1>, <2, 1>), (<2, 1>, <1, 2>), (<2, 1>, <3, 2>), (<3, 1>, <2, 1>),
 (<1, 2>, <2, 2>), (<3, 2>, <2, 2>)]

sage: DA = DiGraph([[2, 1], [3, 2]])
sage: AR = DA.auslander_reiten_quiver()
sage: AR.digraph().edges(labels=False)
[(<1, 1>, <2, 1>), (<2, 1>, <3, 1>), (<2, 1>, <1, 2>), (<3, 1>, <2, 2>),
 (<1, 2>, <2, 2>), (<2, 2>, <1, 3>)]

An example for the type \(D_5\) Dynkin quiver:

sage: DD = DiGraph([[5,3], [4,3], [3,2], [2,1]])
sage: AR = DD.auslander_reiten_quiver()
sage: AR
Auslander-Reiten quiver of a ['D', 5] Dynkin quiver
sage: len(list(DD))
5

An \(E_8\) Dynkin quiver:

sage: DE = DiGraph([[8,7], [7,6], [5,6], [5,3], [3,4], [3,2], [2,1]])
sage: AR = DE.auslander_reiten_quiver()
sage: AR
Auslander-Reiten quiver of a ['E', 8] Dynkin quiver
sage: len(list(AR))
120
sage: len(list(RootSystem(['E', 8]).root_lattice().positive_roots()))
120

The Kronecker quiver:

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: for i in range(1, 5):
....:     for v in D.vertices():
....:         pp = AR(v, i)
....:         pi = AR(v, -i)
....:         print(pp, pp.dimension_vector(), "  ", pi, pi.dimension_vector())
<1, 1> v1 + 2*v2      <1, -1> v1
<2, 1> v2             <2, -1> 2*v1 + v2
<1, 2> 3*v1 + 4*v2    <1, -2> 3*v1 + 2*v2
<2, 2> 2*v1 + 3*v2    <2, -2> 4*v1 + 3*v2
<1, 3> 5*v1 + 6*v2    <1, -3> 5*v1 + 4*v2
<2, 3> 4*v1 + 5*v2    <2, -3> 6*v1 + 5*v2
<1, 4> 7*v1 + 8*v2    <1, -4> 7*v1 + 6*v2
<2, 4> 6*v1 + 7*v2    <2, -4> 8*v1 + 7*v2

class Element(parent, vertex, level)

Bases: Element

A node in the AR quiver.

dimension_vector()

Return the dimension vector of self.

EXAMPLES:

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: node = AR(2, -4)
sage: node.dimension_vector()
8*v1 + 7*v2

inverse_translation()

Return the inverse AR translation of self.

EXAMPLES:

sage: DA = DiGraph([[2,3], [2,1]])
sage: AR = DA.auslander_reiten_quiver()
sage: node = AR(1, 1)
sage: node.inverse_translation()
<1, 2>
sage: node = AR(1, 2)
sage: node.inverse_translation() is None
True

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: AR(2, -1).inverse_translation() is None
True

level()

Return the level of self.

EXAMPLES:

sage: DA = DiGraph([[2,3], [2,1]])
sage: AR = DA.auslander_reiten_quiver()
sage: node = AR(1, 2)
sage: node.level()
2

translation()

Return the AR translation of self.

EXAMPLES:

sage: DA = DiGraph([[4,3], [3,2], [2,1]])
sage: AR = DA.auslander_reiten_quiver()
sage: node = AR(1, 1)
sage: node.translation() is None
True
sage: node = AR(1, 2)
sage: node.translation()
<1, 1>

vertex()

Return the vertex of the quiver corresponding to self.

EXAMPLES:

sage: DA = DiGraph([[2,3], [2,1]])
sage: AR = DA.auslander_reiten_quiver()
sage: node = AR(1, 2)
sage: node.vertex()
1

digraph(with_translations=False)

Return the diagraph of self.

INPUT:

• with_translations – (default: False) if True, then include the arrows corresponding to the translations.

EXAMPLES:

sage: DA = DiGraph([[1,2]])
sage: AR = DA.auslander_reiten_quiver()
sage: G = AR.digraph(); G
Digraph on 3 vertices
sage: G.edges()
[(<1, 1>, <2, 2>, None), (<2, 1>, <1, 1>, None)]
sage: GT = AR.digraph(with_translations=True)
sage: GT.edges()
[(<1, 1>, <2, 2>, None), (<2, 1>, <1, 1>, None), (<2, 2>, <2, 1>, 'ART')]

digraph_postinjectives(max_depth, with_translations=False)

Return the diagraph of postinjectives of self up to max_depth.

EXAMPLES:

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: G = AR.digraph_postinjectives(3)
sage: [node.dimension_vector() for node in G]
[5*v1 + 4*v2, 6*v1 + 5*v2, 3*v1 + 2*v2, 4*v1 + 3*v2, v1, 2*v1 + v2]
sage: AR.digraph_postinjectives(0)
Digraph on 0 vertices

digraph_preprojectives(max_depth, with_translations=False)

Return the diagraph of preprojectives of self up to max_depth.

EXAMPLES:

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: G = AR.digraph_preprojectives(3)
sage: [node.dimension_vector() for node in G]
[v1 + 2*v2, v2, 3*v1 + 4*v2, 2*v1 + 3*v2, 5*v1 + 6*v2, 4*v1 + 5*v2]
sage: AR.digraph_preprojectives(0)
Digraph on 0 vertices

dimension_vectors_of_level(k)

Return a Family of dimension vectors of level k.

EXAMPLES:

sage: DA = DiGraph([[4,3], [2,3], [2,1]])
sage: AR = DA.auslander_reiten_quiver()
sage: AR.dimension_vectors_of_level(1)
{1: v1, 2: v1 + v2 + v3, 3: v3, 4: v3 + v4}
sage: AR.dimension_vectors_of_level(3)
{1: v4, 3: v2}
sage: AR.dimension_vectors_of_level(10)
{}
sage: AR.dimension_vectors_of_level(-1)
 {1: v1 + v2, 2: v2, 3: v2 + v3 + v4, 4: v4}
sage: AR.dimension_vectors_of_level(-2)
{1: v3 + v4, 2: v1 + v2 + v3 + v4, 3: v1 + v2 + v3, 4: v2 + v3}

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: AR.dimension_vectors_of_level(1)
{1: v1 + 2*v2, 2: v2}
sage: AR.dimension_vectors_of_level(3)
{1: 5*v1 + 6*v2, 2: 4*v1 + 5*v2}
sage: AR.dimension_vectors_of_level(-1)
{1: v1, 2: 2*v1 + v2}
sage: AR.dimension_vectors_of_level(-3)
{1: 5*v1 + 4*v2, 2: 6*v1 + 5*v2}

injectives()

Return the injectives of self.

EXAMPLES:

sage: DE = DiGraph([[7,6], [6,5], [5,3], [4,3], [2,3], [1,2]])
sage: AR = DE.auslander_reiten_quiver()
sage: AR.injectives()
Finite family {1: <1, 9>, 2: <2, 9>, 3: <3, 9>, 4: <4, 9>,
               5: <5, 9>, 6: <6, 9>, 7: <7, 9>}

options = Current options for AuslanderReitenQuiver   - latex: node

projectives()

Return the projectives of self.

EXAMPLES:

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: AR = D.auslander_reiten_quiver()
sage: AR.projectives()
Finite family {1: <1, 1>, 2: <2, 1>}

quiver()

Return the quiver defining self.

EXAMPLES:

sage: DE = DiGraph([[7,8], [7,6], [5,6], [3,5], [4,3], [2,3], [1,2]])
sage: AR = DE.auslander_reiten_quiver()
sage: AR.quiver() == DE
True

simples()

Return the simples of self.

EXAMPLES:

sage: DE = DiGraph([[7,8], [7,6], [5,6], [3,5], [4,3], [2,3], [1,2]])
sage: AR = DE.auslander_reiten_quiver()
sage: AR.simples()
Finite family {1: <1, 15>,  2: <1, 14>,  3: <8, 4>,  4: <4, 15>,
               5: <8, 3>,  6: <6, 1>,  7: <7, 15>,  8: <8, 1>}

sage.quivers.ar_quiver.detect_dynkin_quiver(quiver)

Determine if quiver is a finite type Dynkin quiver.

EXAMPLES:

sage: from sage.quivers.ar_quiver import detect_dynkin_quiver
sage: D = DiGraph([[1,2], [2,3], [3, 4], [4,0], ['a','b'], ['b','c'], ['c','d'], ['c','e']])
sage: detect_dynkin_quiver(D)
D5xA5

sage: D = DiGraph([[1,2,'a'], [1,2,'b']], multiedges=True)
sage: detect_dynkin_quiver(D) is None
True
sage: D = DiGraph([[1, 2], [2, 3], [1, 3]])
sage: detect_dynkin_quiver(D) is None
True
sage: D = DiGraph([[1,2], [1,3], [1,4], [1,5]])
sage: detect_dynkin_quiver(D) is None
True
sage: D = DiGraph([[1,2], [2,3], [2,4], [4,5], [6,4]])
sage: detect_dynkin_quiver(D) is None
True
sage: D = DiGraph([[1,2], [2,3], [3,4], [4,5], [5,6], [6,7], [7,8], [8,9], [0,3]])
sage: detect_dynkin_quiver(D) is None
True
sage: D = DiGraph([[1,2], [2,3], [3,4], [4,5], [5,6], [6,7], [0,4]])
sage: detect_dynkin_quiver(D) is None
True