Cluster algebras¶
This file constructs cluster algebras using the Parent-Element framework. The implementation mainly utilizes structural theorems from [FZ2007].
The key points being used here are these:
- cluster variables are parametrized by their g-vectors; 
- g-vectors (together with c-vectors) provide a self-standing model for the combinatorics behind any cluster algebra; 
- each cluster variable in any cluster algebra can be computed, by the separation of additions formula, from its g-vector and F-polynomial. 
Accordingly this file provides three classes:
ClusterAlgebra, constructed as a subobject of
sage.rings.polynomial.laurent_polynomial_ring.LaurentPolynomialRing_generic,
is the frontend of this implementation. It provides all the algebraic
features (like ring morphisms), it computes cluster variables, it is
responsible for controlling the exploration of the exchange graph and
serves as the repository for all the data recursively computed so far.
In particular, all g-vectors and all F-polynomials of known cluster
variables as well as a mutation path by which they can be obtained
are recorded. In the optic of efficiency, this implementation does not
store directly the exchange graph nor the exchange relations. Both of
these could be added to ClusterAlgebra with minimal effort.
ClusterAlgebraSeed provides the combinatorial backbone
for ClusterAlgebra. It is an auxiliary class and therefore its
instances should not be directly created by the user. Rather it
should be accessed via ClusterAlgebra.current_seed()
and ClusterAlgebra.initial_seed(). The task of performing current
seed mutations is delegated to this class. Seeds are considered equal if
they have the same parent cluster algebra and they can be obtained from
each other by a permutation of their data (i.e. if they coincide as
unlabelled seeds).  Cluster algebras whose initial seeds are equal in the
above sense are not considered equal but are endowed with coercion maps
to each other.  More generally, a cluster algebra is endowed with coercion
maps from any cluster algebra which is obtained by freezing a collection
of initial cluster variables and/or permuting both cluster variables
and coefficients.
ClusterAlgebraElement is a thin wrapper around
sage.rings.polynomial.laurent_polynomial.LaurentPolynomial
providing all the functions specific to cluster variables.
Elements of a cluster algebra with principal coefficients have special methods
and these are grouped in the subclass PrincipalClusterAlgebraElement.
One more remark about this implementation. Instances of
ClusterAlgebra are built by identifying the initial cluster variables
with the generators of ClusterAlgebra.ambient(). In particular, this
forces a specific embedding into the ambient field of rational expressions. In
view of this, although cluster algebras themselves are independent of the
choice of initial seed, ClusterAlgebra.mutate_initial() is forced to
return a different instance of ClusterAlgebra. At the moment there
is no coercion implemented among the two instances but this could in
principle be added to ClusterAlgebra.mutate_initial().
REFERENCES:
AUTHORS:
- Dylan Rupel (2015-06-15): initial version 
- Salvatore Stella (2015-06-15): initial version 
EXAMPLES:
We begin by creating a simple cluster algebra and printing its initial exchange matrix:
sage: A = ClusterAlgebra(['A', 2]); A
A Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring
sage: A.b_matrix()
[ 0  1]
[-1  0]
>>> from sage.all import *
>>> A = ClusterAlgebra(['A', Integer(2)]); A
A Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring
>>> A.b_matrix()
[ 0  1]
[-1  0]
A is of finite type so we can explore all its exchange graph:
sage: A.explore_to_depth(infinity)
>>> from sage.all import *
>>> A.explore_to_depth(infinity)
and get all its g-vectors, F-polynomials, and cluster variables:
sage: sorted(A.g_vectors_so_far())
[(-1, 0), (-1, 1), (0, -1), (0, 1), (1, 0)]
sage: sorted(A.F_polynomials_so_far(), key=str)
[1, 1, u0 + 1, u0*u1 + u0 + 1, u1 + 1]
sage: sorted(A.cluster_variables_so_far(), key=str)
[(x0 + 1)/x1, (x0 + x1 + 1)/(x0*x1), (x1 + 1)/x0, x0, x1]
>>> from sage.all import *
>>> sorted(A.g_vectors_so_far())
[(-1, 0), (-1, 1), (0, -1), (0, 1), (1, 0)]
>>> sorted(A.F_polynomials_so_far(), key=str)
[1, 1, u0 + 1, u0*u1 + u0 + 1, u1 + 1]
>>> sorted(A.cluster_variables_so_far(), key=str)
[(x0 + 1)/x1, (x0 + x1 + 1)/(x0*x1), (x1 + 1)/x0, x0, x1]
Simple operations among cluster variables behave as expected:
sage: s = A.cluster_variable((0, -1)); s
(x0 + 1)/x1
sage: t = A.cluster_variable((-1, 1)); t
(x1 + 1)/x0
sage: t + s
(x0^2 + x1^2 + x0 + x1)/(x0*x1)
sage: _.parent() == A
True
sage: t - s
(-x0^2 + x1^2 - x0 + x1)/(x0*x1)
sage: _.parent() == A
True
sage: t*s
(x0*x1 + x0 + x1 + 1)/(x0*x1)
sage: _.parent() == A
True
sage: t/s
(x1^2 + x1)/(x0^2 + x0)
sage: _.parent() == A
False
>>> from sage.all import *
>>> s = A.cluster_variable((Integer(0), -Integer(1))); s
(x0 + 1)/x1
>>> t = A.cluster_variable((-Integer(1), Integer(1))); t
(x1 + 1)/x0
>>> t + s
(x0^2 + x1^2 + x0 + x1)/(x0*x1)
>>> _.parent() == A
True
>>> t - s
(-x0^2 + x1^2 - x0 + x1)/(x0*x1)
>>> _.parent() == A
True
>>> t*s
(x0*x1 + x0 + x1 + 1)/(x0*x1)
>>> _.parent() == A
True
>>> t/s
(x1^2 + x1)/(x0^2 + x0)
>>> _.parent() == A
False
Division is not guaranteed to yield an element of A so it returns an
element of A.ambient().fraction_field() instead:
sage: (t/s).parent() == A.ambient().fraction_field()
True
>>> from sage.all import *
>>> (t/s).parent() == A.ambient().fraction_field()
True
We can compute denominator vectors of any element of A:
sage: (t*s).d_vector()
(1, 1)
>>> from sage.all import *
>>> (t*s).d_vector()
(1, 1)
Since we are in rank 2 and we do not have coefficients we can compute the greedy element associated to any denominator vector:
sage: A.rank() == 2 and A.coefficients() == ()
True
sage: A.greedy_element((1, 1))
(x0 + x1 + 1)/(x0*x1)
sage: _ == t*s
False
>>> from sage.all import *
>>> A.rank() == Integer(2) and A.coefficients() == ()
True
>>> A.greedy_element((Integer(1), Integer(1)))
(x0 + x1 + 1)/(x0*x1)
>>> _ == t*s
False
not surprising since there is no cluster in A containing
both t and s:
sage: seeds = A.seeds(mutating_F=false)
sage: [ S for S in seeds if (0, -1) in S and (-1, 1) in S ]
[]
>>> from sage.all import *
>>> seeds = A.seeds(mutating_F=false)
>>> [ S for S in seeds if (Integer(0), -Integer(1)) in S and (-Integer(1), Integer(1)) in S ]
[]
indeed:
sage: A.greedy_element((1, 1)) == A.cluster_variable((-1, 0))
True
>>> from sage.all import *
>>> A.greedy_element((Integer(1), Integer(1))) == A.cluster_variable((-Integer(1), Integer(0)))
True
Disabling F-polynomials in the computation just done was redundant because we already explored the whole exchange graph before. Though in different circumstances it could have saved us considerable time.
g-vectors and F-polynomials can be computed from elements of A only if
A has principal coefficients at the initial seed:
sage: (t*s).g_vector()
Traceback (most recent call last):
...
AttributeError: 'ClusterAlgebra_with_category.element_class' object has no attribute 'g_vector'...
sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True)
sage: A.explore_to_depth(infinity)
sage: s = A.cluster_variable((0, -1)); s
(x0*y1 + 1)/x1
sage: t = A.cluster_variable((-1, 1)); t
(x1 + y0)/x0
sage: (t*s).g_vector()
(-1, 0)
sage: (t*s).F_polynomial()
u0*u1 + u0 + u1 + 1
sage: (t*s).is_homogeneous()
True
sage: (t+s).is_homogeneous()
False
sage: (t+s).homogeneous_components()
{(-1, 1): (x1 + y0)/x0, (0, -1): (x0*y1 + 1)/x1}
>>> from sage.all import *
>>> (t*s).g_vector()
Traceback (most recent call last):
...
AttributeError: 'ClusterAlgebra_with_category.element_class' object has no attribute 'g_vector'...
>>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True)
>>> A.explore_to_depth(infinity)
>>> s = A.cluster_variable((Integer(0), -Integer(1))); s
(x0*y1 + 1)/x1
>>> t = A.cluster_variable((-Integer(1), Integer(1))); t
(x1 + y0)/x0
>>> (t*s).g_vector()
(-1, 0)
>>> (t*s).F_polynomial()
u0*u1 + u0 + u1 + 1
>>> (t*s).is_homogeneous()
True
>>> (t+s).is_homogeneous()
False
>>> (t+s).homogeneous_components()
{(-1, 1): (x1 + y0)/x0, (0, -1): (x0*y1 + 1)/x1}
Each cluster algebra is endowed with a reference to a current seed; it could be useful to assign a name to it:
sage: A = ClusterAlgebra(['F', 4])
sage: len(A.g_vectors_so_far())
4
sage: A.current_seed()
The initial seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring
sage: A.current_seed() == A.initial_seed()
True
sage: S = A.current_seed()
sage: S.b_matrix()
[ 0  1  0  0]
[-1  0 -1  0]
[ 0  2  0  1]
[ 0  0 -1  0]
sage: S.g_matrix()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: S.cluster_variables()
[x0, x1, x2, x3]
>>> from sage.all import *
>>> A = ClusterAlgebra(['F', Integer(4)])
>>> len(A.g_vectors_so_far())
4
>>> A.current_seed()
The initial seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring
>>> A.current_seed() == A.initial_seed()
True
>>> S = A.current_seed()
>>> S.b_matrix()
[ 0  1  0  0]
[-1  0 -1  0]
[ 0  2  0  1]
[ 0  0 -1  0]
>>> S.g_matrix()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
>>> S.cluster_variables()
[x0, x1, x2, x3]
and use S to walk around the exchange graph of A:
sage: S.mutate(0); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating in direction 0
sage: S.b_matrix()
[ 0 -1  0  0]
[ 1  0 -1  0]
[ 0  2  0  1]
[ 0  0 -1  0]
sage: S.g_matrix()
[-1  0  0  0]
[ 1  1  0  0]
[ 0  0  1  0]
[ 0  0  0  1]
sage: S.cluster_variables()
[(x1 + 1)/x0, x1, x2, x3]
sage: S.mutate('sinks'); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating along the sequence [0, 2]
sage: S.mutate([2, 3, 2, 1, 0]); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating along the sequence [0, 3, 2, 1, 0]
sage: S.g_vectors()
[(0, 1, -2, 0), (-1, 2, -2, 0), (0, 1, -1, 0), (0, 0, 0, -1)]
sage: S.cluster_variable(3)
(x2 + 1)/x3
>>> from sage.all import *
>>> S.mutate(Integer(0)); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating in direction 0
>>> S.b_matrix()
[ 0 -1  0  0]
[ 1  0 -1  0]
[ 0  2  0  1]
[ 0  0 -1  0]
>>> S.g_matrix()
[-1  0  0  0]
[ 1  1  0  0]
[ 0  0  1  0]
[ 0  0  0  1]
>>> S.cluster_variables()
[(x1 + 1)/x0, x1, x2, x3]
>>> S.mutate('sinks'); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating along the sequence [0, 2]
>>> S.mutate([Integer(2), Integer(3), Integer(2), Integer(1), Integer(0)]); S
The seed of a Cluster Algebra with cluster variables x0, x1, x2, x3
 and no coefficients over Integer Ring obtained from the initial
 by mutating along the sequence [0, 3, 2, 1, 0]
>>> S.g_vectors()
[(0, 1, -2, 0), (-1, 2, -2, 0), (0, 1, -1, 0), (0, 0, 0, -1)]
>>> S.cluster_variable(Integer(3))
(x2 + 1)/x3
Walking around by mutating S updates the information stored in A:
sage: len(A.g_vectors_so_far())
10
sage: A.current_seed().path_from_initial_seed()
[0, 3, 2, 1, 0]
sage: A.current_seed() == S
True
>>> from sage.all import *
>>> len(A.g_vectors_so_far())
10
>>> A.current_seed().path_from_initial_seed()
[0, 3, 2, 1, 0]
>>> A.current_seed() == S
True
Starting from A.initial_seed() still records data in A but does not
update A.current_seed():
sage: S1 = A.initial_seed()
sage: S1.mutate([2, 1, 3])
sage: len(A.g_vectors_so_far())
11
sage: S1 == A.current_seed()
False
>>> from sage.all import *
>>> S1 = A.initial_seed()
>>> S1.mutate([Integer(2), Integer(1), Integer(3)])
>>> len(A.g_vectors_so_far())
11
>>> S1 == A.current_seed()
False
Since ClusterAlgebra inherits from UniqueRepresentation,
computed data is shared across instances:
sage: A1 = ClusterAlgebra(['F', 4])
sage: A1 is A
True
sage: len(A1.g_vectors_so_far())
11
>>> from sage.all import *
>>> A1 = ClusterAlgebra(['F', Integer(4)])
>>> A1 is A
True
>>> len(A1.g_vectors_so_far())
11
It can be useful, at times to forget all computed data. Because of
UniqueRepresentation this cannot be achieved by simply creating a
new instance; instead it has to be manually triggered by:
sage: A.clear_computed_data()
sage: len(A.g_vectors_so_far())
4
>>> from sage.all import *
>>> A.clear_computed_data()
>>> len(A.g_vectors_so_far())
4
Given a cluster algebra A we may be looking for a specific cluster
variable:
sage: A = ClusterAlgebra(['E', 8, 1])
sage: v = (-1, 1, -1, 1, -1, 1, 0, 0, 1)
sage: A.find_g_vector(v, depth=2)
sage: seq = A.find_g_vector(v); seq  # random
[0, 1, 2, 4, 3]
sage: v in A.initial_seed().mutate(seq, inplace=False).g_vectors()
True
>>> from sage.all import *
>>> A = ClusterAlgebra(['E', Integer(8), Integer(1)])
>>> v = (-Integer(1), Integer(1), -Integer(1), Integer(1), -Integer(1), Integer(1), Integer(0), Integer(0), Integer(1))
>>> A.find_g_vector(v, depth=Integer(2))
>>> seq = A.find_g_vector(v); seq  # random
[0, 1, 2, 4, 3]
>>> v in A.initial_seed().mutate(seq, inplace=False).g_vectors()
True
This also performs mutations of F-polynomials:
sage: A.F_polynomial((-1, 1, -1, 1, -1, 1, 0, 0, 1))
u0*u1*u2*u3*u4 + u0*u1*u2*u4 + u0*u2*u3*u4 + u0*u1*u2 + u0*u2*u4
 + u2*u3*u4 + u0*u2 + u0*u4 + u2*u4 + u0 + u2 + u4 + 1
>>> from sage.all import *
>>> A.F_polynomial((-Integer(1), Integer(1), -Integer(1), Integer(1), -Integer(1), Integer(1), Integer(0), Integer(0), Integer(1)))
u0*u1*u2*u3*u4 + u0*u1*u2*u4 + u0*u2*u3*u4 + u0*u1*u2 + u0*u2*u4
 + u2*u3*u4 + u0*u2 + u0*u4 + u2*u4 + u0 + u2 + u4 + 1
which might not be a good idea in algebras that are too big. One workaround is to first disable F-polynomials and then recompute only the desired mutations:
sage: # long time
sage: A.reset_exploring_iterator(mutating_F=False)
sage: v = (-1, 1, -2, 2, -1, 1, -1, 1, 1)
sage: seq = A.find_g_vector(v); seq  # random
[1, 0, 2, 6, 5, 4, 3, 8, 1]
sage: S = A.initial_seed().mutate(seq, inplace=False)
sage: v in S.g_vectors()
True
sage: A.current_seed().mutate(seq)
sage: A.F_polynomial((-1, 1, -2, 2, -1, 1, -1, 1, 1))
u0*u1^2*u2^2*u3*u4*u5*u6*u8 +
...
2*u2 + u4 + u6 + 1
>>> from sage.all import *
>>> # long time
>>> A.reset_exploring_iterator(mutating_F=False)
>>> v = (-Integer(1), Integer(1), -Integer(2), Integer(2), -Integer(1), Integer(1), -Integer(1), Integer(1), Integer(1))
>>> seq = A.find_g_vector(v); seq  # random
[1, 0, 2, 6, 5, 4, 3, 8, 1]
>>> S = A.initial_seed().mutate(seq, inplace=False)
>>> v in S.g_vectors()
True
>>> A.current_seed().mutate(seq)
>>> A.F_polynomial((-Integer(1), Integer(1), -Integer(2), Integer(2), -Integer(1), Integer(1), -Integer(1), Integer(1), Integer(1)))
u0*u1^2*u2^2*u3*u4*u5*u6*u8 +
...
2*u2 + u4 + u6 + 1
We can manually freeze cluster variables and get coercions in between the two algebras:
sage: A = ClusterAlgebra(['F', 4]); A
A Cluster Algebra with cluster variables x0, x1, x2, x3 and no coefficients
 over Integer Ring
sage: A1 = ClusterAlgebra(A.b_matrix().matrix_from_columns([0, 1, 2]), coefficient_prefix='x'); A1
A Cluster Algebra with cluster variables x0, x1, x2 and coefficient x3
 over Integer Ring
sage: A.has_coerce_map_from(A1)
True
>>> from sage.all import *
>>> A = ClusterAlgebra(['F', Integer(4)]); A
A Cluster Algebra with cluster variables x0, x1, x2, x3 and no coefficients
 over Integer Ring
>>> A1 = ClusterAlgebra(A.b_matrix().matrix_from_columns([Integer(0), Integer(1), Integer(2)]), coefficient_prefix='x'); A1
A Cluster Algebra with cluster variables x0, x1, x2 and coefficient x3
 over Integer Ring
>>> A.has_coerce_map_from(A1)
True
and we also have an immersion of A.base() into A and of A
into A.ambient():
sage: A.has_coerce_map_from(A.base())
True
sage: A.ambient().has_coerce_map_from(A)
True
>>> from sage.all import *
>>> A.has_coerce_map_from(A.base())
True
>>> A.ambient().has_coerce_map_from(A)
True
but there is currently no coercion in between algebras obtained by mutating at the initial seed:
sage: A1 = A.mutate_initial(0); A1
A Cluster Algebra with cluster variables x4, x1, x2, x3 and no coefficients
 over Integer Ring
sage: A.b_matrix() == A1.b_matrix()
False
sage: [X.has_coerce_map_from(Y) for X, Y in [(A, A1), (A1, A)]]
[False, False]
>>> from sage.all import *
>>> A1 = A.mutate_initial(Integer(0)); A1
A Cluster Algebra with cluster variables x4, x1, x2, x3 and no coefficients
 over Integer Ring
>>> A.b_matrix() == A1.b_matrix()
False
>>> [X.has_coerce_map_from(Y) for X, Y in [(A, A1), (A1, A)]]
[False, False]
- class sage.algebras.cluster_algebra.ClusterAlgebra(B, **kwargs)[source]¶
- Bases: - Parent,- UniqueRepresentation- A Cluster Algebra. - INPUT: - data– some data defining a cluster algebra; it can be anything that can be parsed by- ClusterQuiver
- scalars– a ring (default: \(\ZZ\)); the scalars over which the cluster algebra is defined
- cluster_variable_prefix– string (default:- 'x'); it needs to be a valid variable name
- cluster_variable_names– list of strings; each element needs to be a valid variable name; supersedes- cluster_variable_prefix
- coefficient_prefix– string (default:- 'y'); it needs to be a valid variable name
- coefficient_names– list of strings; each element needs to be a valid variable name; supersedes- cluster_variable_prefix
- principal_coefficients– boolean (default:- False); supersedes any coefficient defined by- data
 - ALGORITHM: - The implementation is mainly based on [FZ2007] and [NZ2012]. - EXAMPLES: - sage: B = matrix([(0, 1, 0, 0), (-1, 0, -1, 0), (0, 1, 0, 1), (0, 0, -2, 0), (-1, 0, 0, 0), (0, -1, 0, 0)]) sage: A = ClusterAlgebra(B); A A Cluster Algebra with cluster variables x0, x1, x2, x3 and coefficients y0, y1 over Integer Ring sage: A.gens() (x0, x1, x2, x3, y0, y1) sage: A = ClusterAlgebra(['A', 2]); A A Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True); A.gens() (x0, x1, y0, y1) sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True, coefficient_prefix='x'); A.gens() (x0, x1, x2, x3) sage: A = ClusterAlgebra(['A', 3], principal_coefficients=True, cluster_variable_names=['a', 'b', 'c']); A.gens() (a, b, c, y0, y1, y2) sage: A = ClusterAlgebra(['A', 3], principal_coefficients=True, cluster_variable_names=['a', 'b']) Traceback (most recent call last): ... ValueError: cluster_variable_names should be an iterable of 3 valid variable names sage: A = ClusterAlgebra(['A', 3], principal_coefficients=True, coefficient_names=['a', 'b', 'c']); A.gens() (x0, x1, x2, a, b, c) sage: A = ClusterAlgebra(['A', 3], principal_coefficients=True, coefficient_names=['a', 'b']) Traceback (most recent call last): ... ValueError: coefficient_names should be an iterable of 3 valid variable names - >>> from sage.all import * >>> B = matrix([(Integer(0), Integer(1), Integer(0), Integer(0)), (-Integer(1), Integer(0), -Integer(1), Integer(0)), (Integer(0), Integer(1), Integer(0), Integer(1)), (Integer(0), Integer(0), -Integer(2), Integer(0)), (-Integer(1), Integer(0), Integer(0), Integer(0)), (Integer(0), -Integer(1), Integer(0), Integer(0))]) >>> A = ClusterAlgebra(B); A A Cluster Algebra with cluster variables x0, x1, x2, x3 and coefficients y0, y1 over Integer Ring >>> A.gens() (x0, x1, x2, x3, y0, y1) >>> A = ClusterAlgebra(['A', Integer(2)]); A A Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True); A.gens() (x0, x1, y0, y1) >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True, coefficient_prefix='x'); A.gens() (x0, x1, x2, x3) >>> A = ClusterAlgebra(['A', Integer(3)], principal_coefficients=True, cluster_variable_names=['a', 'b', 'c']); A.gens() (a, b, c, y0, y1, y2) >>> A = ClusterAlgebra(['A', Integer(3)], principal_coefficients=True, cluster_variable_names=['a', 'b']) Traceback (most recent call last): ... ValueError: cluster_variable_names should be an iterable of 3 valid variable names >>> A = ClusterAlgebra(['A', Integer(3)], principal_coefficients=True, coefficient_names=['a', 'b', 'c']); A.gens() (x0, x1, x2, a, b, c) >>> A = ClusterAlgebra(['A', Integer(3)], principal_coefficients=True, coefficient_names=['a', 'b']) Traceback (most recent call last): ... ValueError: coefficient_names should be an iterable of 3 valid variable names - F_polynomial(g_vector)[source]¶
- Return the F-polynomial with g-vector - g_vectorif it has been found.- INPUT: - g_vector– tuple; the g-vector of the F-polynomial to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.F_polynomial((-1, 1)) Traceback (most recent call last): ... KeyError: 'the g-vector (-1, 1) has not been found yet' sage: A.initial_seed().mutate(0, mutating_F=False) sage: A.F_polynomial((-1, 1)) Traceback (most recent call last): ... KeyError: 'the F-polynomial with g-vector (-1, 1) has not been computed yet; you can compute it by mutating from the initial seed along the sequence [0]' sage: A.initial_seed().mutate(0) sage: A.F_polynomial((-1, 1)) u0 + 1 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.F_polynomial((-Integer(1), Integer(1))) Traceback (most recent call last): ... KeyError: 'the g-vector (-1, 1) has not been found yet' >>> A.initial_seed().mutate(Integer(0), mutating_F=False) >>> A.F_polynomial((-Integer(1), Integer(1))) Traceback (most recent call last): ... KeyError: 'the F-polynomial with g-vector (-1, 1) has not been computed yet; you can compute it by mutating from the initial seed along the sequence [0]' >>> A.initial_seed().mutate(Integer(0)) >>> A.F_polynomial((-Integer(1), Integer(1))) u0 + 1 
 - F_polynomials()[source]¶
- Return an iterator producing all the F_polynomials of - self.- ALGORITHM: - This method does not use the caching framework provided by - self, but recomputes all the F-polynomials from scratch. On the other hand it stores the results so that other methods like- F_polynomials_so_far()can access them afterwards.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: len(list(A.F_polynomials())) 9 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> len(list(A.F_polynomials())) 9 
 - F_polynomials_so_far()[source]¶
- Return a list of the F-polynomials encountered so far. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.current_seed().mutate(0) sage: sorted(A.F_polynomials_so_far(), key=str) [1, 1, u0 + 1] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.current_seed().mutate(Integer(0)) >>> sorted(A.F_polynomials_so_far(), key=str) [1, 1, u0 + 1] 
 - ambient()[source]¶
- Return the Laurent polynomial ring containing - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.ambient() Multivariate Laurent Polynomial Ring in x0, x1, y0, y1 over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.ambient() Multivariate Laurent Polynomial Ring in x0, x1, y0, y1 over Integer Ring 
 - b_matrix()[source]¶
- Return the initial exchange matrix of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.b_matrix() [ 0 1] [-1 0] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.b_matrix() [ 0 1] [-1 0] 
 - clear_computed_data()[source]¶
- Clear the cache of computed g-vectors and F-polynomials and reset both the current seed and the exploring iterator. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: sorted(A.g_vectors_so_far()) [(0, 1), (1, 0)] sage: A.current_seed().mutate([1, 0]) sage: sorted(A.g_vectors_so_far()) [(-1, 0), (0, -1), (0, 1), (1, 0)] sage: A.clear_computed_data() sage: sorted(A.g_vectors_so_far()) [(0, 1), (1, 0)] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> sorted(A.g_vectors_so_far()) [(0, 1), (1, 0)] >>> A.current_seed().mutate([Integer(1), Integer(0)]) >>> sorted(A.g_vectors_so_far()) [(-1, 0), (0, -1), (0, 1), (1, 0)] >>> A.clear_computed_data() >>> sorted(A.g_vectors_so_far()) [(0, 1), (1, 0)] 
 - cluster_fan(depth=+Infinity)[source]¶
- Return the cluster fan (the fan of g-vectors) of - self.- INPUT: - depth– positive integer or infinity (default:- infinity); the maximum depth at which to compute
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.cluster_fan() # needs sage.geometry.polyhedron Rational polyhedral fan in 2-d lattice N - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.cluster_fan() # needs sage.geometry.polyhedron Rational polyhedral fan in 2-d lattice N 
 - cluster_variable(g_vector)[source]¶
- Return the cluster variable with g-vector - g_vectorif it has been found.- INPUT: - g_vector– tuple; the g-vector of the cluster variable to return
 - ALGORITHM: - This function computes cluster variables from their g-vectors and F-polynomials using the “separation of additions” formula of Theorem 3.7 in [FZ2007]. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.initial_seed().mutate(0) sage: A.cluster_variable((-1, 1)) (x1 + 1)/x0 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.initial_seed().mutate(Integer(0)) >>> A.cluster_variable((-Integer(1), Integer(1))) (x1 + 1)/x0 
 - cluster_variables()[source]¶
- Return an iterator producing all the cluster variables of - self.- ALGORITHM: - This method does not use the caching framework provided by - self, but recomputes all the cluster variables from scratch. On the other hand it stores the results so that other methods like- cluster_variables_so_far()can access them afterwards.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: len(list(A.cluster_variables())) 9 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> len(list(A.cluster_variables())) 9 
 - cluster_variables_so_far()[source]¶
- Return a list of the cluster variables encountered so far. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.current_seed().mutate(0) sage: sorted(A.cluster_variables_so_far(), key=str) [(x1 + 1)/x0, x0, x1] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.current_seed().mutate(Integer(0)) >>> sorted(A.cluster_variables_so_far(), key=str) [(x1 + 1)/x0, x0, x1] 
 - coefficient(j)[source]¶
- Return the - j-th coefficient of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the coefficient to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.coefficient(0) y0 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.coefficient(Integer(0)) y0 
 - coefficient_names()[source]¶
- Return the list of coefficient names. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: A.coefficient_names() () sage: A1 = ClusterAlgebra(['B', 2], principal_coefficients=True) sage: A1.coefficient_names() ('y0', 'y1') sage: A2 = ClusterAlgebra(['C', 3], principal_coefficients=True, coefficient_prefix='x') sage: A2.coefficient_names() ('x3', 'x4', 'x5') - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> A.coefficient_names() () >>> A1 = ClusterAlgebra(['B', Integer(2)], principal_coefficients=True) >>> A1.coefficient_names() ('y0', 'y1') >>> A2 = ClusterAlgebra(['C', Integer(3)], principal_coefficients=True, coefficient_prefix='x') >>> A2.coefficient_names() ('x3', 'x4', 'x5') 
 - coefficients()[source]¶
- Return the list of coefficients of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.coefficients() (y0, y1) sage: A1 = ClusterAlgebra(['B', 2]) sage: A1.coefficients() () - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.coefficients() (y0, y1) >>> A1 = ClusterAlgebra(['B', Integer(2)]) >>> A1.coefficients() () 
 - contains_seed(seed)[source]¶
- Test if - seedis a seed of- self.- INPUT: - seed– a- ClusterAlgebraSeed
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True); A A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring sage: S = copy(A.current_seed()) sage: A.contains_seed(S) True - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True); A A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring >>> S = copy(A.current_seed()) >>> A.contains_seed(S) True 
 - coxeter_element()[source]¶
- Return the Coxeter element associated to the initial exchange matrix, if acyclic. - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,1,1],[-1,0,1],[-1,-1,0]])) sage: A.coxeter_element() [0, 1, 2] - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),Integer(1)],[-Integer(1),Integer(0),Integer(1)],[-Integer(1),-Integer(1),Integer(0)]])) >>> A.coxeter_element() [0, 1, 2] - Raise an error if the initial exchange matrix is not acyclic: - sage: A = ClusterAlgebra(matrix([[0,1,-1],[-1,0,1],[1,-1,0]])) sage: A.coxeter_element() Traceback (most recent call last): ... ValueError: the initial exchange matrix is not acyclic - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),-Integer(1)],[-Integer(1),Integer(0),Integer(1)],[Integer(1),-Integer(1),Integer(0)]])) >>> A.coxeter_element() Traceback (most recent call last): ... ValueError: the initial exchange matrix is not acyclic 
 - current_seed()[source]¶
- Return the current seed of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.current_seed() The initial seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.current_seed() The initial seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring 
 - d_vector_to_g_vector(d)[source]¶
- Return the g-vector of an element of - selfhaving d-vector- d- INPUT: - d– the d-vector
 - ALGORITHM: - This method implements the piecewise-linear map \(\\nu_c\) introduced in Section 9.1 of [ReSt2020]. - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,1,1],[-1,0,1],[-1,-1,0]])) sage: A.d_vector_to_g_vector((1,0,-1)) (-1, 1, 2) - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),Integer(1)],[-Integer(1),Integer(0),Integer(1)],[-Integer(1),-Integer(1),Integer(0)]])) >>> A.d_vector_to_g_vector((Integer(1),Integer(0),-Integer(1))) (-1, 1, 2) 
 - euler_matrix()[source]¶
- Return the Euler matrix associated to - self.- ALGORITHM: - This method returns the matrix of the bilinear form defined in Equation (2.1) of [ReSt2020] . - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,1,1],[-1,0,1],[-1,-1,0]])) sage: A.euler_matrix() [ 1 0 0] [-1 1 0] [-1 -1 1] - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),Integer(1)],[-Integer(1),Integer(0),Integer(1)],[-Integer(1),-Integer(1),Integer(0)]])) >>> A.euler_matrix() [ 1 0 0] [-1 1 0] [-1 -1 1] - Raise an error if the initial exchange matrix is not acyclic: - sage: A = ClusterAlgebra(matrix([[0,1,-1],[-1,0,1],[1,-1,0]])) sage: A.euler_matrix() Traceback (most recent call last): ... ValueError: the initial exchange matrix is not acyclic - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),-Integer(1)],[-Integer(1),Integer(0),Integer(1)],[Integer(1),-Integer(1),Integer(0)]])) >>> A.euler_matrix() Traceback (most recent call last): ... ValueError: the initial exchange matrix is not acyclic 
 - explore_to_depth(depth)[source]¶
- Explore the exchange graph of - selfup to distance- depthfrom the initial seed.- INPUT: - depth– positive integer or infinity; the maximum depth at which to stop searching
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 4]) sage: A.explore_to_depth(infinity) sage: len(A.g_vectors_so_far()) 14 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(4)]) >>> A.explore_to_depth(infinity) >>> len(A.g_vectors_so_far()) 14 
 - find_g_vector(g_vector, depth=+Infinity)[source]¶
- Return a mutation sequence to obtain a seed containing the g-vector - g_vectorfrom the initial seed.- INPUT: - g_vector– tuple; the g-vector to find
- depth– positive integer or infinity (default:- infinity); the maximum distance from- self.current_seedto reach
 - OUTPUT: - This function returns a list of integers if it can find - g_vector, otherwise it returns- None. If the exploring iterator stops, it means that the algebra is of finite type and- g_vectoris not the g-vector of any cluster variable. In this case the function resets the iterator and raises an error.- EXAMPLES: - sage: A = ClusterAlgebra(['G', 2], principal_coefficients=True) sage: A.clear_computed_data() sage: A.find_g_vector((-2, 3), depth=2) sage: A.find_g_vector((-2, 3), depth=3) [0, 1, 0] sage: A.find_g_vector((1, 1), depth=3) sage: A.find_g_vector((1, 1), depth=4) Traceback (most recent call last): ... ValueError: (1, 1) is not the g-vector of any cluster variable of a Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['G', Integer(2)], principal_coefficients=True) >>> A.clear_computed_data() >>> A.find_g_vector((-Integer(2), Integer(3)), depth=Integer(2)) >>> A.find_g_vector((-Integer(2), Integer(3)), depth=Integer(3)) [0, 1, 0] >>> A.find_g_vector((Integer(1), Integer(1)), depth=Integer(3)) >>> A.find_g_vector((Integer(1), Integer(1)), depth=Integer(4)) Traceback (most recent call last): ... ValueError: (1, 1) is not the g-vector of any cluster variable of a Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring 
 - g_vector_to_d_vector(g)[source]¶
- Return the d-vector of an element of - selfhaving g-vector- g- INPUT: - g– the g-vector
 - ALGORITHM: - This method implements the inverse of the piecewise-linear map \(\\nu_c\) introduced in Section 9.1 of [ReSt2020]. - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,1,1],[-1,0,1],[-1,-1,0]])) sage: A.g_vector_to_d_vector((-1,1,2)) (1, 0, -1) - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),Integer(1)],[-Integer(1),Integer(0),Integer(1)],[-Integer(1),-Integer(1),Integer(0)]])) >>> A.g_vector_to_d_vector((-Integer(1),Integer(1),Integer(2))) (1, 0, -1) 
 - g_vectors(mutating_F=True)[source]¶
- Return an iterator producing all the g-vectors of - self.- INPUT: - mutating_F– boolean (default:- True); whether to compute F-polynomials; disable this for speed considerations
 - ALGORITHM: - This method does not use the caching framework provided by - self, but recomputes all the g-vectors from scratch. On the other hand it stores the results so that other methods like- g_vectors_so_far()can access them afterwards.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: len(list(A.g_vectors())) 9 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> len(list(A.g_vectors())) 9 
 - g_vectors_so_far()[source]¶
- Return a list of the g-vectors of cluster variables encountered so far. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.current_seed().mutate(0) sage: sorted(A.g_vectors_so_far()) [(-1, 1), (0, 1), (1, 0)] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.current_seed().mutate(Integer(0)) >>> sorted(A.g_vectors_so_far()) [(-1, 1), (0, 1), (1, 0)] 
 - gens()[source]¶
- Return the list of initial cluster variables and coefficients of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.gens() (x0, x1, y0, y1) sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True, coefficient_prefix='x') sage: A.gens() (x0, x1, x2, x3) - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.gens() (x0, x1, y0, y1) >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True, coefficient_prefix='x') >>> A.gens() (x0, x1, x2, x3) 
 - greedy_element(d_vector)[source]¶
- Return the greedy element with denominator vector - d_vector.- INPUT: - d_vector– tuple of 2 integers; the denominator vector of the element to compute
 - ALGORITHM: - This implements greedy elements of a rank 2 cluster algebra using Equation (1.5) from [LLZ2014]. - EXAMPLES: - sage: A = ClusterAlgebra(['A', [1, 1], 1]) sage: A.greedy_element((1, 1)) (x0^2 + x1^2 + 1)/(x0*x1) - >>> from sage.all import * >>> A = ClusterAlgebra(['A', [Integer(1), Integer(1)], Integer(1)]) >>> A.greedy_element((Integer(1), Integer(1))) (x0^2 + x1^2 + 1)/(x0*x1) 
 - initial_cluster_variable(j)[source]¶
- Return the - j-th initial cluster variable of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the cluster variable to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.initial_cluster_variable(0) x0 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.initial_cluster_variable(Integer(0)) x0 
 - initial_cluster_variable_names()[source]¶
- Return the list of initial cluster variable names. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.initial_cluster_variable_names() ('x0', 'x1') sage: A1 = ClusterAlgebra(['B', 2], cluster_variable_prefix='a') sage: A1.initial_cluster_variable_names() ('a0', 'a1') - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.initial_cluster_variable_names() ('x0', 'x1') >>> A1 = ClusterAlgebra(['B', Integer(2)], cluster_variable_prefix='a') >>> A1.initial_cluster_variable_names() ('a0', 'a1') 
 - initial_cluster_variables()[source]¶
- Return the list of initial cluster variables of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: A.initial_cluster_variables() (x0, x1) - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> A.initial_cluster_variables() (x0, x1) 
 - initial_seed()[source]¶
- Return the initial seed of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.initial_seed() The initial seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.initial_seed() The initial seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring 
 - is_acyclic()[source]¶
- Return - Trueif the exchange matrix in the initial seed is acyclic,- Falseotherwise.- EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,1,1],[-1,0,1],[-1,-1,0]])) sage: A.is_acyclic() True sage: A = ClusterAlgebra(matrix([[0,1,-1],[-1,0,1],[1,-1,0]])) sage: A.is_acyclic() False - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),Integer(1)],[-Integer(1),Integer(0),Integer(1)],[-Integer(1),-Integer(1),Integer(0)]])) >>> A.is_acyclic() True >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(1),-Integer(1)],[-Integer(1),Integer(0),Integer(1)],[Integer(1),-Integer(1),Integer(0)]])) >>> A.is_acyclic() False 
 - lift(x)[source]¶
- Return - xas an element of- ambient().- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: x = A.cluster_variable((1, 0)) sage: A.lift(x).parent() Multivariate Laurent Polynomial Ring in x0, x1, y0, y1 over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> x = A.cluster_variable((Integer(1), Integer(0))) >>> A.lift(x).parent() Multivariate Laurent Polynomial Ring in x0, x1, y0, y1 over Integer Ring 
 - lower_bound()[source]¶
- Return the lower bound associated to - self.- EXAMPLES: - sage: A = ClusterAlgebra(['F', 4]) sage: A.lower_bound() Traceback (most recent call last): ... NotImplementedError: not implemented yet - >>> from sage.all import * >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.lower_bound() Traceback (most recent call last): ... NotImplementedError: not implemented yet 
 - mutate_initial(direction, **kwargs)[source]¶
- Return the cluster algebra obtained by mutating - selfat the initial seed.- Warning - This method is significantly slower than - ClusterAlgebraSeed.mutate(). It is therefore advisable to use the latter for exploration purposes.- INPUT: - direction– in which direction(s) to mutate, it can be:- an integer in - range(self.rank())to mutate in one direction only
- an iterable of such integers to mutate along a sequence 
- a string “sinks” or “sources” to mutate at all sinks or sources simultaneously 
 
- mutating_F– boolean (default:- True); whether to compute F-polynomials while mutating
 - Note - While knowing F-polynomials is essential to computing cluster variables, the process of mutating them is quite slow. If you care only about combinatorial data like g-vectors and c-vectors, setting - mutating_F=Falseyields significant benefits in terms of speed.- ALGORITHM: - This function computes data for the new algebra from known data for the old algebra using Equation (4.2) from [NZ2012] for g-vectors, and Equation (6.21) from [FZ2007] for F-polynomials. The exponent \(h\) in the formula for F-polynomials is - -min(0, old_g_vect[k])due to [NZ2012] Proposition 4.2.- EXAMPLES: - sage: A = ClusterAlgebra(['F', 4]) sage: A.explore_to_depth(infinity) sage: B = A.b_matrix() sage: B.mutate(0) sage: A1 = ClusterAlgebra(B) sage: A1.explore_to_depth(infinity) sage: A2 = A1.mutate_initial(0) sage: A2._F_poly_dict == A._F_poly_dict True - >>> from sage.all import * >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.explore_to_depth(infinity) >>> B = A.b_matrix() >>> B.mutate(Integer(0)) >>> A1 = ClusterAlgebra(B) >>> A1.explore_to_depth(infinity) >>> A2 = A1.mutate_initial(Integer(0)) >>> A2._F_poly_dict == A._F_poly_dict True - Check that we did not mess up the original algebra because of - UniqueRepresentation:- sage: A = ClusterAlgebra(['A',2]) sage: A.mutate_initial(0) is A False - >>> from sage.all import * >>> A = ClusterAlgebra(['A',Integer(2)]) >>> A.mutate_initial(Integer(0)) is A False - A faster example without recomputing F-polynomials: - sage: A = ClusterAlgebra(matrix([[0,5],[-5,0]])) sage: A.mutate_initial([0,1]*10, mutating_F=False) A Cluster Algebra with cluster variables x20, x21 and no coefficients over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),Integer(5)],[-Integer(5),Integer(0)]])) >>> A.mutate_initial([Integer(0),Integer(1)]*Integer(10), mutating_F=False) A Cluster Algebra with cluster variables x20, x21 and no coefficients over Integer Ring - Check that Issue #28176 is fixed: - sage: A = ClusterAlgebra( matrix(5,[0,1,-1,1,-1]), cluster_variable_names=['p13'], coefficient_names=['p12','p23','p34','p41']); A A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Integer Ring sage: A.mutate_initial(0) A Cluster Algebra with cluster variable x0 and coefficients p12, p23, p34, p41 over Integer Ring sage: A1 = ClusterAlgebra(['A',[2,1],1]) sage: A2 = A1.mutate_initial([0,1,0]) sage: len(A2.g_vectors_so_far()) == len(A2.F_polynomials_so_far()) True sage: all(parent(f) == A2._U for f in A2.F_polynomials_so_far()) True sage: A2.find_g_vector((0,0,1)) == [] True - >>> from sage.all import * >>> A = ClusterAlgebra( matrix(Integer(5),[Integer(0),Integer(1),-Integer(1),Integer(1),-Integer(1)]), cluster_variable_names=['p13'], coefficient_names=['p12','p23','p34','p41']); A A Cluster Algebra with cluster variable p13 and coefficients p12, p23, p34, p41 over Integer Ring >>> A.mutate_initial(Integer(0)) A Cluster Algebra with cluster variable x0 and coefficients p12, p23, p34, p41 over Integer Ring >>> A1 = ClusterAlgebra(['A',[Integer(2),Integer(1)],Integer(1)]) >>> A2 = A1.mutate_initial([Integer(0),Integer(1),Integer(0)]) >>> len(A2.g_vectors_so_far()) == len(A2.F_polynomials_so_far()) True >>> all(parent(f) == A2._U for f in A2.F_polynomials_so_far()) True >>> A2.find_g_vector((Integer(0),Integer(0),Integer(1))) == [] True 
 - rank()[source]¶
- Return the rank of - self, i.e. the number of cluster variables in any seed.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True); A A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring sage: A.rank() 2 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True); A A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring >>> A.rank() 2 
 - reset_current_seed()[source]¶
- Reset the value reported by - current_seed()to- initial_seed().- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: A.current_seed().mutate([1, 0]) sage: A.current_seed() == A.initial_seed() False sage: A.reset_current_seed() sage: A.current_seed() == A.initial_seed() True - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> A.current_seed().mutate([Integer(1), Integer(0)]) >>> A.current_seed() == A.initial_seed() False >>> A.reset_current_seed() >>> A.current_seed() == A.initial_seed() True 
 - reset_exploring_iterator(mutating_F=True)[source]¶
- Reset the iterator used to explore - self.- INPUT: - mutating_F– boolean (default:- True); whether to also compute F-polynomials; disable this for speed considerations
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 4]) sage: A.clear_computed_data() sage: A.reset_exploring_iterator(mutating_F=False) sage: A.explore_to_depth(infinity) sage: len(A.g_vectors_so_far()) 14 sage: len(A.F_polynomials_so_far()) 4 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(4)]) >>> A.clear_computed_data() >>> A.reset_exploring_iterator(mutating_F=False) >>> A.explore_to_depth(infinity) >>> len(A.g_vectors_so_far()) 14 >>> len(A.F_polynomials_so_far()) 4 
 - retract(x)[source]¶
- Return - xas an element of- self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2], principal_coefficients=True) sage: L = A.ambient() sage: x = L.gen(0) sage: A.retract(x).parent() A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)], principal_coefficients=True) >>> L = A.ambient() >>> x = L.gen(Integer(0)) >>> A.retract(x).parent() A Cluster Algebra with cluster variables x0, x1 and coefficients y0, y1 over Integer Ring 
 - scalars()[source]¶
- Return the ring of scalars over which - selfis defined.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.scalars() Integer Ring - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.scalars() Integer Ring 
 - seeds(**kwargs)[source]¶
- Return an iterator running over seeds of - self.- INPUT: - from_current_seed– boolean (default:- False); whether to start the iterator from- current_seed()or- initial_seed()
- mutating_F– boolean (default:- True); whether to compute F-polynomials also; disable this for speed considerations
- allowed_directions– iterable of integers (default:- range(self.rank())); the directions in which to mutate
- depth– positive integer or infinity (default:- infinity); the maximum depth at which to stop searching
- catch_KeyboardInterrupt– boolean (default:- False); whether to catch- KeyboardInterruptand return it rather then raising an exception – this allows the iterator returned by this method to be resumed after being interrupted
 - ALGORITHM: - This function traverses the exchange graph in a breadth-first search. - EXAMPLES: - sage: A = ClusterAlgebra(['A', 4]) sage: A.clear_computed_data() sage: seeds = A.seeds(allowed_directions=[3, 0, 1]) sage: _ = list(seeds) sage: sorted(A.g_vectors_so_far()) [(-1, 0, 0, 0), (-1, 1, 0, 0), (0, -1, 0, 0), (0, 0, 0, -1), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(4)]) >>> A.clear_computed_data() >>> seeds = A.seeds(allowed_directions=[Integer(3), Integer(0), Integer(1)]) >>> _ = list(seeds) >>> sorted(A.g_vectors_so_far()) [(-1, 0, 0, 0), (-1, 1, 0, 0), (0, -1, 0, 0), (0, 0, 0, -1), (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)] 
 - set_current_seed(seed)[source]¶
- Set the value reported by - current_seed()to- seed, if it makes sense.- INPUT: - seed– a- ClusterAlgebraSeed
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: A.clear_computed_data() sage: S = copy(A.current_seed()) sage: S.mutate([0, 1, 0]) sage: A.current_seed() == S False sage: A.set_current_seed(S) sage: A.current_seed() == S True sage: A1 = ClusterAlgebra(['B', 2]) sage: A.set_current_seed(A1.initial_seed()) Traceback (most recent call last): ... ValueError: this is not a seed in this cluster algebra - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> A.clear_computed_data() >>> S = copy(A.current_seed()) >>> S.mutate([Integer(0), Integer(1), Integer(0)]) >>> A.current_seed() == S False >>> A.set_current_seed(S) >>> A.current_seed() == S True >>> A1 = ClusterAlgebra(['B', Integer(2)]) >>> A.set_current_seed(A1.initial_seed()) Traceback (most recent call last): ... ValueError: this is not a seed in this cluster algebra 
 - theta_basis_F_polynomial(g_vector)[source]¶
- Return the F-polynomial of the element of the theta basis of - selfwith g-vector- g_vector.- INPUT: - g_vector– tuple; the g-vector of the F-polynomial to compute
 - Warning - Elements of the theta basis do not satisfy a separation of additions formula. See the implementation of - sage.algebras.cluster_algebra.theta_basis_F_polynomial()for further details.- ALGORITHM: - This method uses the fact that the greedy basis and the theta basis coincide in rank 2 and uses the former defining recursion (Equation (1.5) from [LLZ2014]) to compute. - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,-3],[2,0]]), principal_coefficients=True) sage: A.theta_basis_F_polynomial((-1,-1)) u0^4*u1 + 4*u0^3*u1 + 6*u0^2*u1 + 4*u0*u1 + u0 + u1 + 1 sage: A = ClusterAlgebra(['F', 4]) sage: A.theta_basis_F_polynomial((1, 0, 0, 0)) Traceback (most recent call last): ... NotImplementedError: currently only implemented for cluster algebras of rank 2 - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),-Integer(3)],[Integer(2),Integer(0)]]), principal_coefficients=True) >>> A.theta_basis_F_polynomial((-Integer(1),-Integer(1))) u0^4*u1 + 4*u0^3*u1 + 6*u0^2*u1 + 4*u0*u1 + u0 + u1 + 1 >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.theta_basis_F_polynomial((Integer(1), Integer(0), Integer(0), Integer(0))) Traceback (most recent call last): ... NotImplementedError: currently only implemented for cluster algebras of rank 2 
 - theta_basis_element(g_vector)[source]¶
- Return the element of the theta basis of - selfwith g-vector- g_vector.- INPUT: - g_vector– tuple; the g-vector of the element to compute
 - EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,-3],[2,0]]), principal_coefficients=True) sage: A.theta_basis_element((-1,-1)) (x1^8*y0^4*y1 + 4*x1^6*y0^3*y1 + 6*x1^4*y0^2*y1 + x0^3*x1^2*y0 + 4*x1^2*y0*y1 + x0^3 + y1)/(x0^4*x1) sage: A = ClusterAlgebra(['F', 4]) sage: A.theta_basis_element((1, 0, 0, 0)) Traceback (most recent call last): ... NotImplementedError: currently only implemented for cluster algebras of rank 2 - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),-Integer(3)],[Integer(2),Integer(0)]]), principal_coefficients=True) >>> A.theta_basis_element((-Integer(1),-Integer(1))) (x1^8*y0^4*y1 + 4*x1^6*y0^3*y1 + 6*x1^4*y0^2*y1 + x0^3*x1^2*y0 + 4*x1^2*y0*y1 + x0^3 + y1)/(x0^4*x1) >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.theta_basis_element((Integer(1), Integer(0), Integer(0), Integer(0))) Traceback (most recent call last): ... NotImplementedError: currently only implemented for cluster algebras of rank 2 - Note - Elements of the theta basis correspond with the associated cluster monomial only for appropriate coefficient choices. For example: - sage: A = ClusterAlgebra(matrix([[0,-1],[1,0],[-1,0]])) sage: A.theta_basis_element((-1,0)) (x1 + y0)/(x0*y0) - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),-Integer(1)],[Integer(1),Integer(0)],[-Integer(1),Integer(0)]])) >>> A.theta_basis_element((-Integer(1),Integer(0))) (x1 + y0)/(x0*y0) - while: - sage: _ = A.find_g_vector((-1,0)); sage: A.cluster_variable((-1,0)) (x1 + y0)/x0 - >>> from sage.all import * >>> _ = A.find_g_vector((-Integer(1),Integer(0))); >>> A.cluster_variable((-Integer(1),Integer(0))) (x1 + y0)/x0 - In particular theta basis elements do not satisfy a separation of additions formula. - Warning - Currently only cluster algebras of rank 2 are supported - See also - sage.algebras.cluster_algebra.theta_basis_F_polynomial()
 - upper_bound()[source]¶
- Return the upper bound associated to - self.- EXAMPLES: - sage: A = ClusterAlgebra(['F', 4]) sage: A.upper_bound() Traceback (most recent call last): ... NotImplementedError: not implemented yet - >>> from sage.all import * >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.upper_bound() Traceback (most recent call last): ... NotImplementedError: not implemented yet 
 - upper_cluster_algebra()[source]¶
- Return the upper cluster algebra associated to - self.- EXAMPLES: - sage: A = ClusterAlgebra(['F', 4]) sage: A.upper_cluster_algebra() Traceback (most recent call last): ... NotImplementedError: not implemented yet - >>> from sage.all import * >>> A = ClusterAlgebra(['F', Integer(4)]) >>> A.upper_cluster_algebra() Traceback (most recent call last): ... NotImplementedError: not implemented yet 
 
- class sage.algebras.cluster_algebra.ClusterAlgebraElement[source]¶
- Bases: - ElementWrapper- An element of a cluster algebra. - d_vector()[source]¶
- Return the denominator vector of - selfas a tuple of integers.- EXAMPLES: - sage: A = ClusterAlgebra(['F', 4], principal_coefficients=True) sage: A.current_seed().mutate([0, 2, 1]) sage: x = A.cluster_variable((-1, 2, -2, 2)) * A.cluster_variable((0, 0, 0, 1))**2 sage: x.d_vector() (1, 1, 2, -2) - >>> from sage.all import * >>> A = ClusterAlgebra(['F', Integer(4)], principal_coefficients=True) >>> A.current_seed().mutate([Integer(0), Integer(2), Integer(1)]) >>> x = A.cluster_variable((-Integer(1), Integer(2), -Integer(2), Integer(2))) * A.cluster_variable((Integer(0), Integer(0), Integer(0), Integer(1)))**Integer(2) >>> x.d_vector() (1, 1, 2, -2) 
 
- class sage.algebras.cluster_algebra.ClusterAlgebraSeed(B, C, G, parent, **kwargs)[source]¶
- Bases: - SageObject- A seed in a Cluster Algebra. - INPUT: - B– a skew-symmetrizable integer matrix
- C– the matrix of c-vectors of- self
- G– the matrix of g-vectors of- self
- parent–- ClusterAlgebra; the algebra to which the seed belongs
- path– list (default:- []); the mutation sequence from the initial seed of- parentto- self
 - Warning - Seeds should not be created manually: no test is performed to assert that they are built from consistent data nor that they really are seeds of - parent. If you create seeds with inconsistent data all sort of things can go wrong, even- __eq__()is no longer guaranteed to give correct answers. Use at your own risk.- F_polynomial(j)[source]¶
- Return the - j-th F-polynomial of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the F-polynomial to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.F_polynomial(0) 1 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.F_polynomial(Integer(0)) 1 
 - F_polynomials()[source]¶
- Return all the F-polynomials of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.F_polynomials() [1, 1, 1] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.F_polynomials() [1, 1, 1] 
 - b_matrix()[source]¶
- Return the exchange matrix of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] 
 - c_matrix()[source]¶
- Return the matrix whose columns are the c-vectors of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.c_matrix() [1 0 0] [0 1 0] [0 0 1] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.c_matrix() [1 0 0] [0 1 0] [0 0 1] 
 - c_vector(j)[source]¶
- Return the - j-th c-vector of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the c-vector to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.c_vector(0) (1, 0, 0) sage: S.mutate(0) sage: S.c_vector(0) (-1, 0, 0) sage: S.c_vector(1) (1, 1, 0) - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.c_vector(Integer(0)) (1, 0, 0) >>> S.mutate(Integer(0)) >>> S.c_vector(Integer(0)) (-1, 0, 0) >>> S.c_vector(Integer(1)) (1, 1, 0) 
 - c_vectors()[source]¶
- Return all the c-vectors of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.c_vectors() [(1, 0, 0), (0, 1, 0), (0, 0, 1)] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.c_vectors() [(1, 0, 0), (0, 1, 0), (0, 0, 1)] 
 - cluster_variable(j)[source]¶
- Return the - j-th cluster variable of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the cluster variable to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.cluster_variable(0) x0 sage: S.mutate(0) sage: S.cluster_variable(0) (x1 + 1)/x0 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.cluster_variable(Integer(0)) x0 >>> S.mutate(Integer(0)) >>> S.cluster_variable(Integer(0)) (x1 + 1)/x0 
 - cluster_variables()[source]¶
- Return all the cluster variables of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.cluster_variables() [x0, x1, x2] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.cluster_variables() [x0, x1, x2] 
 - depth()[source]¶
- Return the length of a mutation sequence from the initial seed
- of - parent()to- self.
 - Warning - This is the length of the mutation sequence returned by - path_from_initial_seed(), which need not be the shortest possible.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: S1 = A.initial_seed() sage: S1.mutate([0, 1, 0, 1]) sage: S1.depth() 4 sage: S2 = A.initial_seed() sage: S2.mutate(1) sage: S2.depth() 1 sage: S1 == S2 True - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> S1 = A.initial_seed() >>> S1.mutate([Integer(0), Integer(1), Integer(0), Integer(1)]) >>> S1.depth() 4 >>> S2 = A.initial_seed() >>> S2.mutate(Integer(1)) >>> S2.depth() 1 >>> S1 == S2 True 
 - g_matrix()[source]¶
- Return the matrix whose columns are the g-vectors of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.g_matrix() [1 0 0] [0 1 0] [0 0 1] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.g_matrix() [1 0 0] [0 1 0] [0 0 1] 
 - g_vector(j)[source]¶
- Return the - j-th g-vector of- self.- INPUT: - j– integer in- range(self.parent().rank()); the index of the g-vector to return
 - EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.g_vector(0) (1, 0, 0) - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.g_vector(Integer(0)) (1, 0, 0) 
 - g_vectors()[source]¶
- Return all the g-vectors of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 3]) sage: S = A.initial_seed() sage: S.g_vectors() [(1, 0, 0), (0, 1, 0), (0, 0, 1)] - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(3)]) >>> S = A.initial_seed() >>> S.g_vectors() [(1, 0, 0), (0, 1, 0), (0, 0, 1)] 
 - mutate(direction, **kwargs)[source]¶
- Mutate - self.- INPUT: - direction– in which direction(s) to mutate, it can be:- an integer in - range(self.rank())to mutate in one direction only
- an iterable of such integers to mutate along a sequence 
- a string “sinks” or “sources” to mutate at all sinks or sources simultaneously 
 
- inplace– boolean (default:- True); whether to mutate in place or to return a new object
- mutating_F– boolean (default:- True); whether to compute F-polynomials while mutating
 - Note - While knowing F-polynomials is essential to computing cluster variables, the process of mutating them is quite slow. If you care only about combinatorial data like g-vectors and c-vectors, setting - mutating_F=Falseyields significant benefits in terms of speed.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: S = A.initial_seed() sage: S.mutate(0); S The seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring obtained from the initial by mutating in direction 0 sage: S.mutate(5) Traceback (most recent call last): ... ValueError: cannot mutate in direction 5 - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> S = A.initial_seed() >>> S.mutate(Integer(0)); S The seed of a Cluster Algebra with cluster variables x0, x1 and no coefficients over Integer Ring obtained from the initial by mutating in direction 0 >>> S.mutate(Integer(5)) Traceback (most recent call last): ... ValueError: cannot mutate in direction 5 
 - parent()[source]¶
- Return the parent of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['B', 3]) sage: A.current_seed().parent() == A True - >>> from sage.all import * >>> A = ClusterAlgebra(['B', Integer(3)]) >>> A.current_seed().parent() == A True 
 - path_from_initial_seed()[source]¶
- Return a mutation sequence from the initial seed of - parent()to- self.- Warning - This is the path used to compute - selfand it does not have to be the shortest possible.- EXAMPLES: - sage: A = ClusterAlgebra(['A', 2]) sage: S1 = A.initial_seed() sage: S1.mutate([0, 1, 0, 1]) sage: S1.path_from_initial_seed() [0, 1, 0, 1] sage: S2 = A.initial_seed() sage: S2.mutate(1) sage: S2.path_from_initial_seed() [1] sage: S1 == S2 True - >>> from sage.all import * >>> A = ClusterAlgebra(['A', Integer(2)]) >>> S1 = A.initial_seed() >>> S1.mutate([Integer(0), Integer(1), Integer(0), Integer(1)]) >>> S1.path_from_initial_seed() [0, 1, 0, 1] >>> S2 = A.initial_seed() >>> S2.mutate(Integer(1)) >>> S2.path_from_initial_seed() [1] >>> S1 == S2 True 
 
- class sage.algebras.cluster_algebra.PrincipalClusterAlgebraElement[source]¶
- Bases: - ClusterAlgebraElement- An element in a cluster algebra with principle coefficients. - F_polynomial()[source]¶
- Return the F-polynomial of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['B', 2], principal_coefficients=True) sage: S = A.initial_seed() sage: S.mutate([0, 1, 0]) sage: S.cluster_variable(0).F_polynomial() == S.F_polynomial(0) True sage: sum(A.initial_cluster_variables()).F_polynomial() Traceback (most recent call last): ... ValueError: this element does not have a well defined g-vector - >>> from sage.all import * >>> A = ClusterAlgebra(['B', Integer(2)], principal_coefficients=True) >>> S = A.initial_seed() >>> S.mutate([Integer(0), Integer(1), Integer(0)]) >>> S.cluster_variable(Integer(0)).F_polynomial() == S.F_polynomial(Integer(0)) True >>> sum(A.initial_cluster_variables()).F_polynomial() Traceback (most recent call last): ... ValueError: this element does not have a well defined g-vector 
 - g_vector()[source]¶
- Return the g-vector of - self.- EXAMPLES: - sage: A = ClusterAlgebra(['B', 2], principal_coefficients=True) sage: A.cluster_variable((1, 0)).g_vector() == (1, 0) True sage: sum(A.initial_cluster_variables()).g_vector() Traceback (most recent call last): ... ValueError: this element does not have a well defined g-vector - >>> from sage.all import * >>> A = ClusterAlgebra(['B', Integer(2)], principal_coefficients=True) >>> A.cluster_variable((Integer(1), Integer(0))).g_vector() == (Integer(1), Integer(0)) True >>> sum(A.initial_cluster_variables()).g_vector() Traceback (most recent call last): ... ValueError: this element does not have a well defined g-vector 
 - homogeneous_components()[source]¶
- Return a dictionary of the homogeneous components of - self.- OUTPUT: - A dictionary whose keys are homogeneous degrees and whose values are the summands of - selfof the given degree.- EXAMPLES: - sage: A = ClusterAlgebra(['B', 2], principal_coefficients=True) sage: x = A.cluster_variable((1, 0)) + A.cluster_variable((0, 1)) sage: x.homogeneous_components() {(0, 1): x1, (1, 0): x0} - >>> from sage.all import * >>> A = ClusterAlgebra(['B', Integer(2)], principal_coefficients=True) >>> x = A.cluster_variable((Integer(1), Integer(0))) + A.cluster_variable((Integer(0), Integer(1))) >>> x.homogeneous_components() {(0, 1): x1, (1, 0): x0} 
 - is_homogeneous()[source]¶
- Return - Trueif- selfis a homogeneous element of- self.parent().- EXAMPLES: - sage: A = ClusterAlgebra(['B', 2], principal_coefficients=True) sage: A.cluster_variable((1, 0)).is_homogeneous() True sage: x = A.cluster_variable((1, 0)) + A.cluster_variable((0, 1)) sage: x.is_homogeneous() False - >>> from sage.all import * >>> A = ClusterAlgebra(['B', Integer(2)], principal_coefficients=True) >>> A.cluster_variable((Integer(1), Integer(0))).is_homogeneous() True >>> x = A.cluster_variable((Integer(1), Integer(0))) + A.cluster_variable((Integer(0), Integer(1))) >>> x.is_homogeneous() False 
 - theta_basis_decomposition()[source]¶
- Return the decomposition of - selfin the theta basis.- OUTPUT: - A dictionary whose keys are the g-vectors and whose values are the coefficients in the decomposition of - selfin the theta basis.- EXAMPLES: - sage: A = ClusterAlgebra(matrix([[0,-2],[3,0]]), principal_coefficients=True) sage: f = (A.theta_basis_element((1,0)) + A.theta_basis_element((0,1)))**2 + A.coefficient(1)* A.theta_basis_element((1,1)) sage: decomposition = f.theta_basis_decomposition() sage: sum(decomposition[g] * A.theta_basis_element(g) for g in decomposition) == f True sage: f = A.theta_basis_element((4,-4))*A.theta_basis_element((1,-1)) sage: decomposition = f.theta_basis_decomposition() sage: sum(decomposition[g] * A.theta_basis_element(g) for g in decomposition) == f True - >>> from sage.all import * >>> A = ClusterAlgebra(matrix([[Integer(0),-Integer(2)],[Integer(3),Integer(0)]]), principal_coefficients=True) >>> f = (A.theta_basis_element((Integer(1),Integer(0))) + A.theta_basis_element((Integer(0),Integer(1))))**Integer(2) + A.coefficient(Integer(1))* A.theta_basis_element((Integer(1),Integer(1))) >>> decomposition = f.theta_basis_decomposition() >>> sum(decomposition[g] * A.theta_basis_element(g) for g in decomposition) == f True >>> f = A.theta_basis_element((Integer(4),-Integer(4)))*A.theta_basis_element((Integer(1),-Integer(1))) >>> decomposition = f.theta_basis_decomposition() >>> sum(decomposition[g] * A.theta_basis_element(g) for g in decomposition) == f True