ClusterSeed¶
A cluster seed is a pair with
being a skew-symmetrizable
-matrix
and with
being an
-tuple of independent elements in the field of rational functions in
variables.
For the compendium on the cluster algebra and quiver package see Arxiv 1102.4844.
AUTHORS:
- Gregg Musiker
- Christian Stump
See also
For mutation types of cluster seeds, see sage.combinat.cluster_algebra_quiver.quiver_mutation_type.QuiverMutationType()
. Cluster seeds are closely related to sage.combinat.cluster_algebra_quiver.quiver.ClusterQuiver()
.
-
class
sage.combinat.cluster_algebra_quiver.cluster_seed.
ClusterSeed
(data, frozen=None, is_principal=None)¶ Bases:
sage.structure.sage_object.SageObject
The cluster seed associated to an exchange matrix.
INPUT:
data
– can be any of the following:* QuiverMutationType * str - a string representing a QuiverMutationType or a common quiver type (see Examples) * ClusterQuiver * Matrix - a skew-symmetrizable matrix * DiGraph - must be the input data for a quiver * List of edges - must be the edge list of a digraph for a quiver
EXAMPLES:
sage: S = ClusterSeed(['A',5]); S A seed for a cluster algebra of rank 5 of type ['A', 5] sage: S = ClusterSeed(['A',[2,5],1]); S A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1] sage: T = ClusterSeed( S ); T A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1] sage: T = ClusterSeed( S._M ); T A seed for a cluster algebra of rank 7 sage: T = ClusterSeed( S.quiver()._digraph ); T A seed for a cluster algebra of rank 7 sage: T = ClusterSeed( S.quiver()._digraph.edges() ); T A seed for a cluster algebra of rank 7 sage: S = ClusterSeed(['B',2]); S A seed for a cluster algebra of rank 2 of type ['B', 2] sage: S = ClusterSeed(['C',2]); S A seed for a cluster algebra of rank 2 of type ['B', 2] sage: S = ClusterSeed(['A', [5,0],1]); S A seed for a cluster algebra of rank 5 of type ['D', 5] sage: S = ClusterSeed(['GR',[3,7]]); S A seed for a cluster algebra of rank 6 of type ['E', 6] sage: S = ClusterSeed(['F', 4, [2,1]]); S A seed for a cluster algebra of rank 6 of type ['F', 4, [1, 2]]
-
b_matrix
()¶ Returns the
-matrix of
self
.EXAMPLES:
sage: ClusterSeed(['A',4]).b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] sage: ClusterSeed(['B',4]).b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -2 0] sage: ClusterSeed(['D',4]).b_matrix() [ 0 1 0 0] [-1 0 -1 -1] [ 0 1 0 0] [ 0 1 0 0] sage: ClusterSeed(QuiverMutationType([['A',2],['B',2]])).b_matrix() [ 0 1 0 0] [-1 0 0 0] [ 0 0 0 1] [ 0 0 -2 0]
-
b_matrix_class
(depth=+Infinity, up_to_equivalence=True)¶ Returns all
-matrices in the mutation class of
self
.INPUT:
depth
– (default:infinity) integer or infinity, only seeds with distance at most depth from self are returnedup_to_equivalence
– (default: True) if True, only ‘B’-matrices up to equivalence are considered.
EXAMPLES:
- for examples see
b_matrix_class_iter()
TESTS:
sage: A = ClusterSeed(['A',3]).b_matrix_class() sage: A = ClusterSeed(['A',[2,1],1]).b_matrix_class()
-
b_matrix_class_iter
(depth=+Infinity, up_to_equivalence=True)¶ Returns an iterator through all
-matrices in the mutation class of
self
.INPUT:
depth
– (default:infinity) integer or infinity, only seeds with distance at most depth from self are returnedup_to_equivalence
– (default: True) if True, only ‘B’-matrices up to equivalence are considered.
EXAMPLES:
A standard finite type example:
sage: S = ClusterSeed(['A',4]) sage: it = S.b_matrix_class_iter() sage: for T in it: print T [ 0 0 0 1] [ 0 0 1 1] [ 0 -1 0 0] [-1 -1 0 0] [ 0 0 0 1] [ 0 0 1 0] [ 0 -1 0 1] [-1 0 -1 0] [ 0 0 1 1] [ 0 0 0 -1] [-1 0 0 0] [-1 1 0 0] [ 0 0 0 1] [ 0 0 -1 1] [ 0 1 0 -1] [-1 -1 1 0] [ 0 0 0 1] [ 0 0 -1 0] [ 0 1 0 -1] [-1 0 1 0] [ 0 0 0 -1] [ 0 0 -1 1] [ 0 1 0 -1] [ 1 -1 1 0]
A finite type example with given depth:
sage: it = S.b_matrix_class_iter(depth=1) sage: for T in it: print T [ 0 0 0 1] [ 0 0 1 1] [ 0 -1 0 0] [-1 -1 0 0] [ 0 0 0 1] [ 0 0 1 0] [ 0 -1 0 1] [-1 0 -1 0] [ 0 0 1 1] [ 0 0 0 -1] [-1 0 0 0] [-1 1 0 0]
Finite type example not considered up to equivalence:
sage: S = ClusterSeed(['A',3]) sage: it = S.b_matrix_class_iter(up_to_equivalence=False) sage: for T in it: print T [ 0 1 0] [-1 0 -1] [ 0 1 0] [ 0 1 0] [-1 0 1] [ 0 -1 0] [ 0 -1 0] [ 1 0 1] [ 0 -1 0] [ 0 -1 0] [ 1 0 -1] [ 0 1 0] [ 0 -1 1] [ 1 0 -1] [-1 1 0] [ 0 1 -1] [-1 0 1] [ 1 -1 0] [ 0 0 1] [ 0 0 -1] [-1 1 0] [ 0 -1 1] [ 1 0 0] [-1 0 0] [ 0 0 -1] [ 0 0 1] [ 1 -1 0] [ 0 1 -1] [-1 0 0] [ 1 0 0] [ 0 1 1] [-1 0 0] [-1 0 0] [ 0 -1 -1] [ 1 0 0] [ 1 0 0] [ 0 0 -1] [ 0 0 -1] [ 1 1 0] [ 0 0 1] [ 0 0 1] [-1 -1 0]
Infinite (but finite mutation) type example:
sage: S = ClusterSeed(['A',[1,2],1]) sage: it = S.b_matrix_class_iter() sage: for T in it: print T [ 0 1 1] [-1 0 1] [-1 -1 0] [ 0 -2 1] [ 2 0 -1] [-1 1 0]
Infinite mutation type example:
sage: S = ClusterSeed(['E',10]) sage: it = S.b_matrix_class_iter(depth=3) sage: len ( [T for T in it] ) 266
-
c_matrix
(ignore_coefficients=False)¶ Returns all c-vectors of
self
.Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: S.c_matrix() [ 1 0 0] [ 0 0 -1] [ 0 -1 0]
-
c_vector
(k, ignore_coefficients=False)¶ Returns the
k
-th c-vector ofself
. It is obtained as thek
-th column vector of the bottom part of theB
-matrix ofself
.Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: [ S.c_vector(k) for k in range(3) ] [(1, 0, 0), (0, 0, -1), (0, -1, 0)] sage: S = ClusterSeed(Matrix([[0,1],[-1,0],[1,0],[-1,1]])); S A seed for a cluster algebra of rank 2 with 2 frozen variables sage: S.c_vector(0) Traceback (most recent call last): ... ValueError: No principal coefficients initialized. Use principal_extension, or ignore_coefficients to ignore this. sage: S.c_vector(0,ignore_coefficients=True) (1, -1)
-
cluster
()¶ Returns the cluster of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.cluster() [x0, x1, x2] sage: S.mutate(1) sage: S.cluster() [x0, (x0*x2 + 1)/x1, x2] sage: S.mutate(2) sage: S.cluster() [x0, (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] sage: S.mutate([2,1]) sage: S.cluster() [x0, x1, x2]
-
cluster_class
(depth=+Infinity, show_depth=False, up_to_equivalence=True)¶ Returns the cluster class of
self
with respect to certain constraints.INPUT:
depth
– (default: infinity) integer, only seeds with distance at most depth from self are returnedreturn_depth
– (default False) - if True, ignored if depth is set; returns the depth of the mutation class, i.e., the maximal distance from self of an element in the mutation classup_to_equivalence
– (default: True) if True, only clusters up to equivalence are considered.
EXAMPLES:
- for examples see
cluster_class_iter()
TESTS:
sage: A = ClusterSeed(['A',3]).cluster_class()
-
cluster_class_iter
(depth=+Infinity, show_depth=False, up_to_equivalence=True)¶ Returns an iterator through all clusters in the mutation class of
self
.INPUT:
depth
– (default: infinity) integer or infinity, only seeds with distance at most depth from self are returnedshow_depth
– (default False) - if True, ignored if depth is set; returns the depth of the mutation class, i.e., the maximal distance from self of an element in the mutation classup_to_equivalence
– (default: True) if True, only clusters up to equivalence are considered.
EXAMPLES:
A standard finite type example:
sage: S = ClusterSeed(['A',3]) sage: it = S.cluster_class_iter() sage: for T in it: print T [x0, x1, x2] [x0, x1, (x1 + 1)/x2] [x0, (x0*x2 + 1)/x1, x2] [(x1 + 1)/x0, x1, x2] [x0, (x0*x2 + x1 + 1)/(x1*x2), (x1 + 1)/x2] [(x1 + 1)/x0, x1, (x1 + 1)/x2] [(x1 + 1)/x0, (x0*x2 + x1 + 1)/(x0*x1), x2] [x0, (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] [(x0*x2 + x1 + 1)/(x0*x1), (x0*x2 + 1)/x1, x2] [(x1 + 1)/x0, (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2), (x1 + 1)/x2] [(x1 + 1)/x0, (x0*x2 + x1 + 1)/(x0*x1), (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2)] [(x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2), (x0*x2 + x1 + 1)/(x1*x2), (x1 + 1)/x2] [(x0*x2 + x1 + 1)/(x0*x1), (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] [(x0*x2 + x1 + 1)/(x1*x2), (x0*x2 + x1 + 1)/(x0*x1), (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2)]
A finite type example with given depth:
sage: it = S.cluster_class_iter(depth=1) sage: for T in it: print T [x0, x1, x2] [x0, x1, (x1 + 1)/x2] [x0, (x0*x2 + 1)/x1, x2] [(x1 + 1)/x0, x1, x2]
A finite type example where the depth is returned while computing:
sage: it = S.cluster_class_iter(show_depth=True) sage: for T in it: print T [x0, x1, x2] Depth: 0 found: 1 Time: ... s [x0, x1, (x1 + 1)/x2] [x0, (x0*x2 + 1)/x1, x2] [(x1 + 1)/x0, x1, x2] Depth: 1 found: 4 Time: ... s [x0, (x0*x2 + x1 + 1)/(x1*x2), (x1 + 1)/x2] [(x1 + 1)/x0, x1, (x1 + 1)/x2] [(x1 + 1)/x0, (x0*x2 + x1 + 1)/(x0*x1), x2] [x0, (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] [(x0*x2 + x1 + 1)/(x0*x1), (x0*x2 + 1)/x1, x2] Depth: 2 found: 9 Time: ... s [(x1 + 1)/x0, (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2), (x1 + 1)/x2] [(x1 + 1)/x0, (x0*x2 + x1 + 1)/(x0*x1), (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2)] [(x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2), (x0*x2 + x1 + 1)/(x1*x2), (x1 + 1)/x2] [(x0*x2 + x1 + 1)/(x0*x1), (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] Depth: 3 found: 13 Time: ... s [(x0*x2 + x1 + 1)/(x1*x2), (x0*x2 + x1 + 1)/(x0*x1), (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2)] Depth: 4 found: 14 Time: ... s
Finite type examples not considered up to equivalence:
sage: it = S.cluster_class_iter(up_to_equivalence=False) sage: len( [ T for T in it ] ) 84 sage: it = ClusterSeed(['A',2]).cluster_class_iter(up_to_equivalence=False) sage: for T in it: print T [x0, x1] [x0, (x0 + 1)/x1] [(x1 + 1)/x0, x1] [(x1 + 1)/x0, (x0 + x1 + 1)/(x0*x1)] [(x0 + x1 + 1)/(x0*x1), (x0 + 1)/x1] [(x0 + x1 + 1)/(x0*x1), (x1 + 1)/x0] [(x0 + 1)/x1, (x0 + x1 + 1)/(x0*x1)] [x1, (x1 + 1)/x0] [(x0 + 1)/x1, x0] [x1, x0]
Infinite type examples:
sage: S = ClusterSeed(['A',[1,1],1]) sage: it = S.cluster_class_iter() sage: next(it) [x0, x1] sage: next(it) [x0, (x0^2 + 1)/x1] sage: next(it) [(x1^2 + 1)/x0, x1] sage: next(it) [(x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2), (x0^2 + 1)/x1] sage: next(it) [(x1^2 + 1)/x0, (x1^4 + x0^2 + 2*x1^2 + 1)/(x0^2*x1)] sage: it = S.cluster_class_iter(depth=3) sage: for T in it: print T [x0, x1] [x0, (x0^2 + 1)/x1] [(x1^2 + 1)/x0, x1] [(x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2), (x0^2 + 1)/x1] [(x1^2 + 1)/x0, (x1^4 + x0^2 + 2*x1^2 + 1)/(x0^2*x1)] [(x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2), (x0^6 + 3*x0^4 + 2*x0^2*x1^2 + x1^4 + 3*x0^2 + 2*x1^2 + 1)/(x0^2*x1^3)] [(x1^6 + x0^4 + 2*x0^2*x1^2 + 3*x1^4 + 2*x0^2 + 3*x1^2 + 1)/(x0^3*x1^2), (x1^4 + x0^2 + 2*x1^2 + 1)/(x0^2*x1)]
-
cluster_variable
(k)¶ Returns the
-th cluster variable of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.mutate([1,2]) sage: [S.cluster_variable(k) for k in range(3)] [x0, (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)]
-
coefficient
(k)¶ Returns the coefficient of
self
at indexk
.EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: [ S.coefficient(k) for k in range(3) ] [y0, 1/y2, 1/y1]
-
coefficients
()¶ Returns all coefficients of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: S.coefficients() [y0, 1/y2, 1/y1]
-
exchangeable_part
()¶ Returns the restriction to the principal part (i.e. the exchangeable variables) of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',4]) sage: T = ClusterSeed( S.quiver().digraph().edges(), frozen=1 ) sage: T.quiver().digraph().edges() [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))] sage: T.exchangeable_part().quiver().digraph().edges() [(0, 1, (1, -1)), (2, 1, (1, -1))] sage: S2 = S.principal_extension() sage: S3 = S2.principal_extension(ignore_coefficients=True) sage: S2.exchangeable_part() == S3.exchangeable_part() True
-
f_polynomial
(k, ignore_coefficients=False)¶ Returns the
k
-th F-polynomial ofself
. It is obtained from thek
-th cluster variable by setting allto
.
Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: [S.f_polynomial(k) for k in range(3)] [1, y1*y2 + y2 + 1, y1 + 1] sage: S = ClusterSeed(Matrix([[0,1],[-1,0],[1,0],[-1,1]])); S A seed for a cluster algebra of rank 2 with 2 frozen variables sage: T = ClusterSeed(Matrix([[0,1],[-1,0]])).principal_extension(); T A seed for a cluster algebra of rank 2 with principal coefficients sage: S.mutate(0) sage: T.mutate(0) sage: S.f_polynomials() Traceback (most recent call last): ... ValueError: No principal coefficients initialized. Use principal_extension, or ignore_coefficients to ignore this. sage: S.f_polynomials(ignore_coefficients=True) [y0 + y1, 1] sage: T.f_polynomials() [y0 + 1, 1]
-
f_polynomials
(ignore_coefficients=False)¶ Returns all F-polynomials of
self
. These are obtained from the cluster variables by setting all‘s to
.
Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: S.f_polynomials() [1, y1*y2 + y2 + 1, y1 + 1]
-
g_matrix
(ignore_coefficients=False)¶ Returns the matrix of all g-vectors of
self
. This are the degree vectors of the cluster variables after setting all‘s to
.
Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: S.g_matrix() [ 1 0 0] [ 0 0 -1] [ 0 -1 0] sage: S = ClusterSeed(['A',3]) sage: S2 = S.principal_extension() sage: S.mutate([0,1]) sage: S2.mutate([0,1]) sage: S.g_matrix() Traceback (most recent call last): ... ValueError: No principal coefficients initialized. Use principal_extension, or ignore_coefficients to ignore this. sage: S.g_matrix(ignore_coefficients=True) [-1 0 0] [ 1 0 0] [ 0 1 1] sage: S2.g_matrix() [-1 -1 0] [ 1 0 0] [ 0 0 1]
-
g_vector
(k, ignore_coefficients=False)¶ Returns the
k
-th g-vector ofself
. This is the degree vector of thek
-th cluster variable after setting all‘s to
.
Requires principal coefficients, initialized by using principal_extension(), or the user can set ‘ignore_coefficients=True’ to bypass this restriction.
Warning: this method assumes the sign-coherence conjecture and that the input seed is sign-coherent (has an exchange matrix with columns of like signs). Otherwise, computational errors might arise.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1,2]) sage: [ S.g_vector(k) for k in range(3) ] [(1, 0, 0), (0, 0, -1), (0, -1, 0)]
-
greedy
(a1, a2, method='by_recursion')¶ Returns the greedy element
assuming that self is rank two.
The third input can be ‘by_recursion’, ‘by_combinatorics’, or ‘just_numbers’ to specify if the user wants the element computed by the recurrence, combinatorial formula, or wants to set
and
to be one.
See [LeeLiZe] for more details.
EXAMPLES:
sage: S = ClusterSeed(['R2', [3, 3]]) sage: S.greedy(4, 4) (x0^12 + x1^12 + 4*x0^9 + 4*x1^9 + 6*x0^6 + 4*x0^3*x1^3 + 6*x1^6 + 4*x0^3 + 4*x1^3 + 1)/(x0^4*x1^4) sage: S.greedy(4, 4, 'by_combinatorics') (x0^12 + x1^12 + 4*x0^9 + 4*x1^9 + 6*x0^6 + 4*x0^3*x1^3 + 6*x1^6 + 4*x0^3 + 4*x1^3 + 1)/(x0^4*x1^4) sage: S.greedy(4, 4, 'just_numbers') 35 sage: S = ClusterSeed(['R2', [2, 2]]) sage: S.greedy(1, 2) (x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2) sage: S.greedy(1, 2, 'by_combinatorics') (x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2)
REFERENCES:
[LeeLiZe] (1, 2) Lee-Li-Zelevinsky, Greedy elements in rank 2 cluster algebras, Arxiv 1208.2391
-
ground_field
()¶ Returns the ground field of the cluster of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.ground_field() Fraction Field of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
-
interact
(fig_size=1, circular=True)¶ Only in notebook mode. Starts an interactive window for cluster seed mutations.
INPUT:
fig_size
– (default: 1) factor by which the size of the plot is multiplied.circular
– (default: True) if True, the circular plot is chosen, otherwise >>spring<< is used.
TESTS:
sage: S = ClusterSeed(['A',4]) sage: S.interact() # long time 'The interactive mode only runs in the Sage notebook.'
-
is_acyclic
()¶ Returns True iff self is acyclic (i.e., if the underlying quiver is acyclic).
EXAMPLES:
sage: ClusterSeed(['A',4]).is_acyclic() True sage: ClusterSeed(['A',[2,1],1]).is_acyclic() True sage: ClusterSeed([[0,1],[1,2],[2,0]]).is_acyclic() False
-
is_bipartite
(return_bipartition=False)¶ Returns True iff self is bipartite (i.e., if the underlying quiver is bipartite).
INPUT:
- return_bipartition – (default:False) if True, the bipartition is returned in the case of
self
being bipartite.
EXAMPLES:
sage: ClusterSeed(['A',[3,3],1]).is_bipartite() True sage: ClusterSeed(['A',[4,3],1]).is_bipartite() False
- return_bipartition – (default:False) if True, the bipartition is returned in the case of
-
is_finite
()¶ Returns True if
self
is of finite type.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.is_finite() True sage: S = ClusterSeed(['A',[2,2],1]) sage: S.is_finite() False
-
is_mutation_finite
(nr_of_checks=None, return_path=False)¶ Returns True if
self
is of finite mutation type.INPUT:
nr_of_checks
– (default: None) number of mutations applied. Standard is 500*(number of vertices of self).return_path
– (default: False) if True, in case of self not being mutation finite, a path from self to a quiver with an edge label (a,-b) and a*b > 4 is returned.
ALGORITHM:
- A cluster seed is mutation infinite if and only if every
. Thus, we apply random mutations in random directions
WARNING:
- Uses a non-deterministic method by random mutations in various directions.
- In theory, it can return a wrong True.
EXAMPLES:
sage: S = ClusterSeed(['A',10]) sage: S._mutation_type = None sage: S.is_mutation_finite() True sage: S = ClusterSeed([(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(2,9)]) sage: S.is_mutation_finite() False
-
m
()¶ Returns the number of frozen variables of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.n() 3 sage: S.m() 0 sage: S = S.principal_extension() sage: S.m() 3
-
mutate
(sequence, inplace=True)¶ Mutates
self
at a vertex or a sequence of vertices.INPUT:
sequence
– a vertex of self or an iterator of vertices of self.inplace
– (default: True) if False, the result is returned, otherwiseself
is modified.
EXAMPLES:
sage: S = ClusterSeed(['A',4]); S.b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] sage: S.mutate(0); S.b_matrix() [ 0 -1 0 0] [ 1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] sage: T = S.mutate(0, inplace=False); T A seed for a cluster algebra of rank 4 of type ['A', 4] sage: S.mutate(0) sage: S == T True sage: S.mutate([0,1,0]) sage: S.b_matrix() [ 0 -1 1 0] [ 1 0 0 0] [-1 0 0 1] [ 0 0 -1 0] sage: S = ClusterSeed(QuiverMutationType([['A',1],['A',3]])) sage: S.b_matrix() [ 0 0 0 0] [ 0 0 1 0] [ 0 -1 0 -1] [ 0 0 1 0] sage: T = S.mutate(0,inplace=False) sage: S == T False
-
mutation_class
(depth=+Infinity, show_depth=False, return_paths=False, up_to_equivalence=True, only_sink_source=False)¶ Returns the mutation class of
self
with respect to certain constraints.INPUT:
depth
– (default: infinity) integer, only seeds with distance at most depth from self are returned.show_depth
– (default: False) if True, the actual depth of the mutation is shown.return_paths
– (default: False) if True, a shortest path of mutation sequences from self to the given quiver is returned as well.up_to_equivalence
– (default: True) if True, only seeds up to equivalence are considered.sink_source
– (default: False) if True, only mutations at sinks and sources are applied.
EXAMPLES:
- for examples see
mutation_class_iter()
TESTS:
sage: A = ClusterSeed(['A',3]).mutation_class()
-
mutation_class_iter
(depth=+Infinity, show_depth=False, return_paths=False, up_to_equivalence=True, only_sink_source=False)¶ Returns an iterator for the mutation class of
self
with respect to certain constrains.INPUT:
depth
– (default: infinity) integer or infinity, only seeds with distance at mostdepth
fromself
are returned.show_depth
– (default: False) if True, the current depth of the mutation is shown while computing.return_paths
– (default: False) if True, a shortest path of mutations fromself
to the given quiver is returned as well.up_to_equivalence
– (default: True) if True, only one seed up to simultaneous permutation of rows and columns of the exchange matrix is recorded.sink_source
– (default: False) if True, only mutations at sinks and sources are applied.
EXAMPLES:
A standard finite type example:
sage: S = ClusterSeed(['A',3]) sage: it = S.mutation_class_iter() sage: for T in it: print T A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3]
A finite type example with given depth:
sage: it = S.mutation_class_iter(depth=1) sage: for T in it: print T A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3] A seed for a cluster algebra of rank 3 of type ['A', 3]
A finite type example where the depth is shown while computing:
sage: it = S.mutation_class_iter(show_depth=True) sage: for T in it: pass Depth: 0 found: 1 Time: ... s Depth: 1 found: 4 Time: ... s Depth: 2 found: 9 Time: ... s Depth: 3 found: 13 Time: ... s Depth: 4 found: 14 Time: ... s
A finite type example with shortest paths returned:
sage: it = S.mutation_class_iter(return_paths=True) sage: for T in it: print T (A seed for a cluster algebra of rank 3 of type ['A', 3], []) (A seed for a cluster algebra of rank 3 of type ['A', 3], [2]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [1]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [2, 1]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0, 2]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0, 1]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [1, 2]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [1, 0]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0, 2, 1]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0, 1, 2]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [2, 1, 0]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [1, 0, 2]) (A seed for a cluster algebra of rank 3 of type ['A', 3], [0, 1, 2, 0])
Finite type examples not considered up to equivalence:
sage: it = S.mutation_class_iter(up_to_equivalence=False) sage: len( [ T for T in it ] ) 84 sage: it = ClusterSeed(['A',2]).mutation_class_iter(return_paths=True,up_to_equivalence=False) sage: for T in it: print T (A seed for a cluster algebra of rank 2 of type ['A', 2], []) (A seed for a cluster algebra of rank 2 of type ['A', 2], [1]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [0]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [0, 1]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [1, 0]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [1, 0, 1]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [0, 1, 0]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [1, 0, 1, 0]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [0, 1, 0, 1]) (A seed for a cluster algebra of rank 2 of type ['A', 2], [1, 0, 1, 0, 1])
Check that trac ticket #14638 is fixed:
sage: S = ClusterSeed(['E',6]) sage: MC = S.mutation_class(depth=7); len(MC) 534
Infinite type examples:
sage: S = ClusterSeed(['A',[1,1],1]) sage: it = S.mutation_class_iter() sage: next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] sage: next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] sage: next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] sage: next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] sage: it = S.mutation_class_iter(depth=3, return_paths=True) sage: for T in it: print T (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], []) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [1]) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [0]) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [1, 0]) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [0, 1]) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [1, 0, 1]) (A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1], [0, 1, 0])
-
mutation_sequence
(sequence, show_sequence=False, fig_size=1.2, return_output='seed')¶ Returns the seeds obtained by mutating
self
at all vertices insequence
.INPUT:
sequence
– an iterable of vertices of self.show_sequence
– (default: False) if True, a png containing the associated quivers is shown.fig_size
– (default: 1.2) factor by which the size of the plot is multiplied.return_output
– (default: ‘seed’) determines what output is to be returned:* if 'seed', outputs all the cluster seeds obtained by the ``sequence`` of mutations. * if 'matrix', outputs a list of exchange matrices. * if 'var', outputs a list of new cluster variables obtained at each step.
EXAMPLES:
sage: S = ClusterSeed(['A',2]) sage: for T in S.mutation_sequence([0,1,0]): ... print T.b_matrix() [ 0 -1] [ 1 0] [ 0 1] [-1 0] [ 0 -1] [ 1 0] sage: S=ClusterSeed(['A',2]) sage: S.mutation_sequence([0,1,0,1],return_output='var') [(x1 + 1)/x0, (x0 + x1 + 1)/(x0*x1), (x0 + 1)/x1, x0]
-
mutation_type
()¶ Returns the mutation_type of each connected component of
self
, if it can be determined. Otherwise, the mutation type of this component is set to be unknown.The mutation types of the components are ordered by vertex labels.
WARNING:
- All finite types can be detected,
- All affine types can be detected, EXCEPT affine type D (the algorithm is not yet implemented)
- All exceptional types can be detected.
- Might fail to work if it is used within different Sage processes simultaneously (that happend in the doctesting).
EXAMPLES:
finite types:
sage: S = ClusterSeed(['A',5]) sage: S._mutation_type = S._quiver._mutation_type = None sage: S.mutation_type() ['A', 5] sage: S = ClusterSeed([(0,1),(1,2),(2,3),(3,4)]) sage: S.mutation_type() ['A', 5]
affine types:
sage: S = ClusterSeed(['E',8,[1,1]]); S A seed for a cluster algebra of rank 10 of type ['E', 8, [1, 1]] sage: S._mutation_type = S._quiver._mutation_type = None; S A seed for a cluster algebra of rank 10 sage: S.mutation_type() # long time ['E', 8, [1, 1]]
the not yet working affine type D:
sage: S = ClusterSeed(['D',4,1]) sage: S._mutation_type = S._quiver._mutation_type = None sage: S.mutation_type() # todo: not implemented ['D', 4, 1]
the exceptional types:
sage: S = ClusterSeed(['X',6]) sage: S._mutation_type = S._quiver._mutation_type = None sage: S.mutation_type() # long time ['X', 6]
infinite types:
sage: S = ClusterSeed(['GR',[4,9]]) sage: S._mutation_type = S._quiver._mutation_type = None sage: S.mutation_type() 'undetermined infinite mutation type'
-
n
()¶ Returns the number of exchangeable variables of
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.n() 3
-
plot
(circular=False, mark=None, save_pos=False)¶ Returns the plot of the quiver of
self
.INPUT:
circular
– (default:False) if True, the circular plot is chosen, otherwise >>spring<< is used.mark
– (default: None) if set to i, the vertex i is highlighted.save_pos
– (default:False) if True, the positions of the vertices are saved.
EXAMPLES:
sage: S = ClusterSeed(['A',5]) sage: pl = S.plot() sage: pl = S.plot(circular=True)
-
principal_extension
(ignore_coefficients=False)¶ Returns the principal extension of self, yielding a 2n-by-n matrix. Raises an error if the input seed has a non-square exchange matrix, unless ‘ignore_coefficients=True’ is set. In this case, the method instead adds n frozen variables to any previously frozen variables. I.e., the seed obtained by adding a frozen variable to every exchangeable variable of
self
.EXAMPLES:
sage: S = ClusterSeed([[0,1],[1,2],[2,3],[2,4]]); S A seed for a cluster algebra of rank 5 sage: T = S.principal_extension(); T A seed for a cluster algebra of rank 5 with principal coefficients sage: T.b_matrix() [ 0 1 0 0 0] [-1 0 1 0 0] [ 0 -1 0 1 1] [ 0 0 -1 0 0] [ 0 0 -1 0 0] [ 1 0 0 0 0] [ 0 1 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 0 0 0 0 1] sage: T2 = T.principal_extension() Traceback (most recent call last): ... ValueError: The b-matrix is not square. Use ignore_coefficients to ignore this. sage: T2 = T.principal_extension(ignore_coefficients=True); T2.b_matrix() [ 0 1 0 0 0] [-1 0 1 0 0] [ 0 -1 0 1 1] [ 0 0 -1 0 0] [ 0 0 -1 0 0] [ 1 0 0 0 0] [ 0 1 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 0 0 0 0 1] [ 1 0 0 0 0] [ 0 1 0 0 0] [ 0 0 1 0 0] [ 0 0 0 1 0] [ 0 0 0 0 1]
-
quiver
()¶ Returns the quiver associated to
self
.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.quiver() Quiver on 3 vertices of type ['A', 3]
-
reorient
(data)¶ Reorients
self
with respect to the given total order, or with respect to an iterator of ordered pairs.WARNING:
- This operation might change the mutation type of
self
. - Ignores ordered pairs
for which neither
nor
is an edge of
self
.
INPUT:
data
– an iterator defining a total order onself.vertices()
, or an iterator of ordered pairs inself
defining the new orientation of these edges.
EXAMPLES:
sage: S = ClusterSeed(['A',[2,3],1]) sage: S.mutation_type() ['A', [2, 3], 1] sage: S.reorient([(0,1),(2,3)]) sage: S.mutation_type() ['D', 5] sage: S.reorient([(1,0),(2,3)]) sage: S.mutation_type() ['A', [1, 4], 1] sage: S.reorient([0,1,2,3,4]) sage: S.mutation_type() ['A', [1, 4], 1]
- This operation might change the mutation type of
-
reset_cluster
()¶ Resets the cluster of
self
to the initial cluster.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.mutate([1,2,1]) sage: S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] sage: S.reset_cluster() sage: S.cluster() [x0, x1, x2] sage: T = S.principal_extension() sage: T.cluster() [x0, x1, x2] sage: T.mutate([1,2,1]) sage: T.cluster() [x0, (x1*y2 + x0)/x2, (x1*y1*y2 + x0*y1 + x2)/(x1*x2)] sage: T.reset_cluster() sage: T.cluster() [x0, x1, x2]
-
reset_coefficients
()¶ Resets the coefficients of
self
to the frozen variables but keeps the current cluster. Raises an error if the number of frozen variables is different than the number of exchangeable variables.EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] [ 1 0 0] [ 0 1 0] [ 0 0 1] sage: S.mutate([1,2,1]) sage: S.b_matrix() [ 0 1 -1] [-1 0 1] [ 1 -1 0] [ 1 0 0] [ 0 1 -1] [ 0 0 -1] sage: S.reset_coefficients() sage: S.b_matrix() [ 0 1 -1] [-1 0 1] [ 1 -1 0] [ 1 0 0] [ 0 1 0] [ 0 0 1]
-
save_image
(filename, circular=False, mark=None, save_pos=False)¶ Saves the plot of the underlying digraph of the quiver of
self
.INPUT:
filename
– the filename the image is saved to.circular
– (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.mark
– (default: None) if set to i, the vertex i is highlighted.save_pos
– (default:False) if True, the positions of the vertices are saved.
EXAMPLES:
sage: S = ClusterSeed(['F',4,[1,2]]) sage: S.save_image(os.path.join(SAGE_TMP, 'sage.png'))
-
set_cluster
(cluster)¶ Sets the cluster for
self
tocluster
.INPUT:
cluster
– an iterable defining a cluster forself
.
EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: cluster = S.cluster() sage: S.mutate([1,2,1]) sage: S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] sage: S.set_cluster(cluster) sage: S.cluster() [x0, x1, x2]
-
show
(fig_size=1, circular=False, mark=None, save_pos=False)¶ Shows the plot of the quiver of
self
.INPUT:
fig_size
– (default: 1) factor by which the size of the plot is multiplied.circular
– (default: False) if True, the circular plot is chosen, otherwise >>spring<< is used.mark
– (default: None) if set to i, the vertex i is highlighted.save_pos
– (default:False) if True, the positions of the vertices are saved.
TESTS:
sage: S = ClusterSeed(['A',5]) sage: S.show() # long time
-
universal_extension
()¶ Returns the universal extension of
self
.This is the initial seed of the associated cluster algebra with universal coefficients, as defined in section 12 of Arxiv math/0602259.
This method works only if
self
is a bipartite, finite-type seed.Due to some limitations in the current implementation of
CartanType
, we need to construct the set of almost positive coroots by hand. As a consequence their ordering is not the standard one (the rows of the bottom part of the exchange matrix might be a shuffling of those you would expect).EXAMPLES:
sage: S = ClusterSeed(['A',2]) sage: T = S.universal_extension() sage: T.b_matrix() [ 0 1] [-1 0] [-1 0] [ 1 0] [ 1 -1] [ 0 1] [ 0 -1] sage: S = ClusterSeed(['A',3]) sage: T = S.universal_extension() sage: T.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] [-1 0 0] [ 1 0 0] [ 1 -1 0] [ 1 -1 1] [ 0 1 0] [ 0 -1 0] [ 0 -1 1] [ 0 0 -1] [ 0 0 1] sage: S = ClusterSeed(['B',2]) sage: T = S.universal_extension() sage: T.b_matrix() [ 0 1] [-2 0] [-1 0] [ 1 0] [ 1 -1] [ 2 -1] [ 0 1] [ 0 -1]
-
variable_class
(depth=+Infinity, ignore_bipartite_belt=False)¶ Returns all cluster variables in the mutation class of
self
.INPUT:
depth
– (default:infinity) integer, only seeds with distance at most depth from self are returnedignore_bipartite_belt
– (default:False) if True, the algorithms does not use the bipartite belt
EXAMPLES:
- for examples see
variable_class_iter()
TESTS:
sage: A = ClusterSeed(['A',3]).variable_class()
-
variable_class_iter
(depth=+Infinity, ignore_bipartite_belt=False)¶ Returns an iterator for all cluster variables in the mutation class of
self
.INPUT:
depth
– (default:infinity) integer, only seeds with distance at most depth from self are returnedignore_bipartite_belt
– (default:False) if True, the algorithms does not use the bipartite belt
EXAMPLES:
A standard finite type example:
sage: S = ClusterSeed(['A',3]) sage: it = S.variable_class_iter() sage: for T in it: print T x0 x1 x2 (x1 + 1)/x0 (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2) (x1 + 1)/x2 (x0*x2 + x1 + 1)/(x0*x1) (x0*x2 + 1)/x1 (x0*x2 + x1 + 1)/(x1*x2)
Finite type examples with given depth:
sage: it = S.variable_class_iter(depth=1) sage: for T in it: print T Found a bipartite seed - restarting the depth counter at zero and constructing the variable class using its bipartite belt. x0 x1 x2 (x1 + 1)/x0 (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2) (x1 + 1)/x2 (x0*x2 + x1 + 1)/(x0*x1) (x0*x2 + 1)/x1 (x0*x2 + x1 + 1)/(x1*x2)
Note that the notion of depth depends on whether a bipartite seed is found or not, or if it is manually ignored:
sage: it = S.variable_class_iter(depth=1,ignore_bipartite_belt=True) sage: for T in it: print T x0 x1 x2 (x1 + 1)/x2 (x0*x2 + 1)/x1 (x1 + 1)/x0 sage: S.mutate([0,1]) sage: it2 = S.variable_class_iter(depth=1) sage: for T in it2: print T (x1 + 1)/x0 (x0*x2 + x1 + 1)/(x0*x1) x2 (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2) x1 (x0*x2 + 1)/x1
Infinite type examples:
sage: S = ClusterSeed(['A',[1,1],1]) sage: it = S.variable_class_iter(depth=2) sage: for T in it: print T Found a bipartite seed - restarting the depth counter at zero and constructing the variable class using its bipartite belt. x0 x1 (x1^2 + 1)/x0 (x1^4 + x0^2 + 2*x1^2 + 1)/(x0^2*x1) (x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2) (x0^2 + 1)/x1 (x1^6 + x0^4 + 2*x0^2*x1^2 + 3*x1^4 + 2*x0^2 + 3*x1^2 + 1)/(x0^3*x1^2) (x1^8 + x0^6 + 2*x0^4*x1^2 + 3*x0^2*x1^4 + 4*x1^6 + 3*x0^4 + 6*x0^2*x1^2 + 6*x1^4 + 3*x0^2 + 4*x1^2 + 1)/(x0^4*x1^3) (x0^8 + 4*x0^6 + 3*x0^4*x1^2 + 2*x0^2*x1^4 + x1^6 + 6*x0^4 + 6*x0^2*x1^2 + 3*x1^4 + 4*x0^2 + 3*x1^2 + 1)/(x0^3*x1^4) (x0^6 + 3*x0^4 + 2*x0^2*x1^2 + x1^4 + 3*x0^2 + 2*x1^2 + 1)/(x0^2*x1^3)
-
x
(k)¶ Returns the
-th initial cluster variable for the associated cluster seed.
EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: S.mutate([2,1]) sage: S.x(0) x0 sage: S.x(1) x1 sage: S.x(2) x2
-
y
(k)¶ Returns the
-th initial coefficient (frozen variable) for the associated cluster seed.
EXAMPLES:
sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutate([2,1]) sage: S.y(0) y0 sage: S.y(1) y1 sage: S.y(2) y2
-
class
sage.combinat.cluster_algebra_quiver.cluster_seed.
ClusterVariable
(parent, numerator, denominator, coerce=True, reduce=True, mutation_type=None, variable_type=None)¶ Bases:
sage.rings.fraction_field_element.FractionFieldElement
This class is a thin wrapper for cluster variables in cluster seeds.
It provides the extra feature to store if a variable is frozen or not.
the associated positive root:
sage: S = ClusterSeed(['A',3]) sage: for T in S.variable_class_iter(): print T, T.almost_positive_root() x0 -alpha[1] x1 -alpha[2] x2 -alpha[3] (x1 + 1)/x0 alpha[1] (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2) alpha[1] + alpha[2] + alpha[3] (x1 + 1)/x2 alpha[3] (x0*x2 + x1 + 1)/(x0*x1) alpha[1] + alpha[2] (x0*x2 + 1)/x1 alpha[2] (x0*x2 + x1 + 1)/(x1*x2) alpha[2] + alpha[3]
-
almost_positive_root
()¶ Returns the almost positive root associated to
self
ifself
is of finite type.EXAMPLES:
sage: S = ClusterSeed(['A',3]) sage: for T in S.variable_class_iter(): print T, T.almost_positive_root() x0 -alpha[1] x1 -alpha[2] x2 -alpha[3] (x1 + 1)/x0 alpha[1] (x1^2 + x0*x2 + 2*x1 + 1)/(x0*x1*x2) alpha[1] + alpha[2] + alpha[3] (x1 + 1)/x2 alpha[3] (x0*x2 + x1 + 1)/(x0*x1) alpha[1] + alpha[2] (x0*x2 + 1)/x1 alpha[2] (x0*x2 + x1 + 1)/(x1*x2) alpha[2] + alpha[3]
-
sage.combinat.cluster_algebra_quiver.cluster_seed.
PathSubset
(n, m)¶ Encodes a maximal Dyck path from (0,0) to (n,m) (for n >= m >= 0) as a subset of {0,1,2,..., 2n-1}. The encoding is given by indexing horizontal edges by odd numbers and vertical edges by evens.
The horizontal between (i,j) and (i+1,j) is indexed by the odd number 2*i+1. The vertical between (i,j) and (i,j+1) is indexed by the even number 2*j.
EXAMPLES:
sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import PathSubset sage: PathSubset(4,0) {1, 3, 5, 7} sage: PathSubset(4,1) {1, 3, 5, 6, 7} sage: PathSubset(4,2) {1, 2, 3, 5, 6, 7} sage: PathSubset(4,3) {1, 2, 3, 4, 5, 6, 7} sage: PathSubset(4,4) {0, 1, 2, 3, 4, 5, 6, 7}
-
sage.combinat.cluster_algebra_quiver.cluster_seed.
SetToPath
(T)¶ Rearranges the encoding for a maximal Dyck path (as a set) so that it is a list in the proper order of the edges.
EXAMPLES:
sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import PathSubset sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import SetToPath sage: SetToPath(PathSubset(4,0)) [1, 3, 5, 7] sage: SetToPath(PathSubset(4,1)) [1, 3, 5, 7, 6] sage: SetToPath(PathSubset(4,2)) [1, 3, 2, 5, 7, 6] sage: SetToPath(PathSubset(4,3)) [1, 3, 2, 5, 4, 7, 6] sage: SetToPath(PathSubset(4,4)) [1, 0, 3, 2, 5, 4, 7, 6]
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sage.combinat.cluster_algebra_quiver.cluster_seed.
coeff_recurs
(p, q, a1, a2, b, c)¶ Coefficients in Laurent expansion of greedy element, as defined by recursion.
EXAMPLES:
sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import coeff_recurs sage: coeff_recurs(1, 1, 5, 5, 3, 3) 10
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sage.combinat.cluster_algebra_quiver.cluster_seed.
is_LeeLiZel_allowable
(T, n, m, b, c)¶ Check if the subset T contributes to the computation of the greedy element x[m,n] in the rank two (b,c)-cluster algebra.
This uses the conditions of Lee-Li-Zelevinsky’s paper [LeeLiZe].
EXAMPLES:
sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import is_LeeLiZel_allowable sage: is_LeeLiZel_allowable({1,3,2,5,7,6},4,2,6,6) False sage: is_LeeLiZel_allowable({1,2,5},3,3,1,1) True