Helper functions for mutation types of quivers#
This file contains helper functions for detecting the mutation type of a cluster algebra or quiver.
For the compendium on the cluster algebra and quiver package see [MS2011]
AUTHORS:
Gregg Musiker
Christian Stump
- sage.combinat.cluster_algebra_quiver.mutation_type.is_mutation_finite(M, nr_of_checks=None)[source]#
Use a non-deterministic method by random mutations in various directions. Can result in a wrong answer.
Warning
This method modifies the input matrix
M!INPUT:
nr_of_checks– (default:None) number of mutations applied. Standard is 500*(number of vertices of self).
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: from sage.combinat.cluster_algebra_quiver.mutation_type import is_mutation_finite sage: Q = ClusterQuiver(['A',10]) # needs sage.modules sage: M = Q.b_matrix() # needs sage.modules sage: is_mutation_finite(M) # needs sage.modules (True, None) sage: # needs sage.modules sage: Q = ClusterQuiver([(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(2,9)]) sage: M = Q.b_matrix() sage: is_mutation_finite(M) # random (False, [9, 6, 9, 8, 9, 4, 0, 4, 5, 2, 1, 0, 1, 0, 7, 1, 9, 2, 5, 7, 8, 6, 3, 0, 2, 5, 4, 2, 6, 9, 2, 7, 3, 5, 3, 7, 9, 5, 9, 0, 2, 7, 9, 2, 4, 2, 1, 6, 9, 4, 3, 5, 0, 8, 2, 9, 5, 3, 7, 0, 1, 8, 3, 7, 2, 7, 3, 4, 8, 0, 4, 9, 5, 2, 8, 4, 8, 1, 7, 8, 9, 1, 5, 0, 8, 7, 4, 8, 9, 8, 0, 7, 4, 7, 1, 2, 8, 6, 1, 3, 9, 3, 9, 1, 3, 2, 4, 9, 5, 1, 2, 9, 4, 8, 5, 3, 4, 6, 8, 9, 2, 5, 9, 4, 6, 2, 1, 4, 9, 6, 0, 9, 8, 0, 4, 7, 9, 2, 1, 6])
>>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.mutation_type import is_mutation_finite >>> Q = ClusterQuiver(['A',Integer(10)]) # needs sage.modules >>> M = Q.b_matrix() # needs sage.modules >>> is_mutation_finite(M) # needs sage.modules (True, None) >>> # needs sage.modules >>> Q = ClusterQuiver([(Integer(0),Integer(1)),(Integer(1),Integer(2)),(Integer(2),Integer(3)),(Integer(3),Integer(4)),(Integer(4),Integer(5)),(Integer(5),Integer(6)),(Integer(6),Integer(7)),(Integer(7),Integer(8)),(Integer(2),Integer(9))]) >>> M = Q.b_matrix() >>> is_mutation_finite(M) # random (False, [9, 6, 9, 8, 9, 4, 0, 4, 5, 2, 1, 0, 1, 0, 7, 1, 9, 2, 5, 7, 8, 6, 3, 0, 2, 5, 4, 2, 6, 9, 2, 7, 3, 5, 3, 7, 9, 5, 9, 0, 2, 7, 9, 2, 4, 2, 1, 6, 9, 4, 3, 5, 0, 8, 2, 9, 5, 3, 7, 0, 1, 8, 3, 7, 2, 7, 3, 4, 8, 0, 4, 9, 5, 2, 8, 4, 8, 1, 7, 8, 9, 1, 5, 0, 8, 7, 4, 8, 9, 8, 0, 7, 4, 7, 1, 2, 8, 6, 1, 3, 9, 3, 9, 1, 3, 2, 4, 9, 5, 1, 2, 9, 4, 8, 5, 3, 4, 6, 8, 9, 2, 5, 9, 4, 6, 2, 1, 4, 9, 6, 0, 9, 8, 0, 4, 7, 9, 2, 1, 6])
Check that Issue #19495 is fixed:
sage: dg = DiGraph(); dg.add_vertex(0); S = ClusterSeed(dg); S # needs sage.modules A seed for a cluster algebra of rank 1 sage: S.is_mutation_finite() # needs sage.modules True
>>> from sage.all import * >>> dg = DiGraph(); dg.add_vertex(Integer(0)); S = ClusterSeed(dg); S # needs sage.modules A seed for a cluster algebra of rank 1 >>> S.is_mutation_finite() # needs sage.modules True
- sage.combinat.cluster_algebra_quiver.mutation_type.load_data(user=True)[source]#
Load a dict with keys being tuples representing exceptional QuiverMutationTypes, and with values being lists or sets containing all mutation equivalent quivers as dig6 data.
We check
the data stored by the user (unless
user=Falsewas given)and the data installed by the optional package
database_mutation_class.
INPUT:
user– boolean (default:True) whether to look at user data. If not, only consider the optional package.
EXAMPLES:
sage: from sage.combinat.cluster_algebra_quiver.mutation_type import load_data sage: load_data(2) # random - depends on the data the user has stored {('G', 2): [('AO', (((0, 1), (1, -3)),)), ('AO', (((0, 1), (3, -1)),))]}
>>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.mutation_type import load_data >>> load_data(Integer(2)) # random - depends on the data the user has stored {('G', 2): [('AO', (((0, 1), (1, -3)),)), ('AO', (((0, 1), (3, -1)),))]}