ClusterSeed¶
A cluster seed is a pair \((B,\mathbf{x})\) with \(B\) being a skew-symmetrizable \((n+m) \times n\) -matrix and with \(\mathbf{x}\) being an \(n\)-tuple of independent elements in the field of rational functions in \(n\) variables.
For the compendium on the cluster algebra and quiver package see [MS2011].
AUTHORS:
- Gregg Musiker: Initial Version 
- Christian Stump: Initial Version 
- Aram Dermenjian (2015-07-01): Updating ability to not rely solely on clusters 
- Jesse Levitt (2015-07-01): Updating ability to not rely solely on clusters 
REFERENCES:
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=False, user_labels=None, user_labels_prefix='x')[source]¶
- Bases: - SageObject- The cluster seed associated to an exchange matrix. - INPUT: - data– can be any of the following:- * :class:`QuiverMutationType`- str– string representing a- QuiverMutationTypeor a common quiver type (see Examples)
- 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(sort=True)); 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]] sage: S = ClusterSeed(['A',4]); S._use_fpolys True sage: S._use_d_vec True sage: S._use_g_vec True sage: S._use_c_vec True sage: S = ClusterSeed(['A', 4]); S.use_fpolys(False); S._use_fpolys False sage: S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ....: frozen=['c']); S A seed for a cluster algebra of rank 4 with 1 frozen variable sage: S = ClusterSeed(['D', 4], user_labels=[-1, 0, 1, 2]); S A seed for a cluster algebra of rank 4 of type ['D', 4] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(5)]); S A seed for a cluster algebra of rank 5 of type ['A', 5] >>> S = ClusterSeed(['A',[Integer(2),Integer(5)],Integer(1)]); S A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1] >>> T = ClusterSeed(S); T A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1] >>> T = ClusterSeed(S._M); T A seed for a cluster algebra of rank 7 >>> T = ClusterSeed(S.quiver()._digraph); T A seed for a cluster algebra of rank 7 >>> T = ClusterSeed(S.quiver()._digraph.edges(sort=True)); T A seed for a cluster algebra of rank 7 >>> S = ClusterSeed(['B',Integer(2)]); S A seed for a cluster algebra of rank 2 of type ['B', 2] >>> S = ClusterSeed(['C',Integer(2)]); S A seed for a cluster algebra of rank 2 of type ['B', 2] >>> S = ClusterSeed(['A', [Integer(5),Integer(0)],Integer(1)]); S A seed for a cluster algebra of rank 5 of type ['D', 5] >>> S = ClusterSeed(['GR',[Integer(3),Integer(7)]]); S A seed for a cluster algebra of rank 6 of type ['E', 6] >>> S = ClusterSeed(['F', Integer(4), [Integer(2),Integer(1)]]); S A seed for a cluster algebra of rank 6 of type ['F', 4, [1, 2]] >>> S = ClusterSeed(['A',Integer(4)]); S._use_fpolys True >>> S._use_d_vec True >>> S._use_g_vec True >>> S._use_c_vec True >>> S = ClusterSeed(['A', Integer(4)]); S.use_fpolys(False); S._use_fpolys False >>> S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ... frozen=['c']); S A seed for a cluster algebra of rank 4 with 1 frozen variable >>> S = ClusterSeed(['D', Integer(4)], user_labels=[-Integer(1), Integer(0), Integer(1), Integer(2)]); S A seed for a cluster algebra of rank 4 of type ['D', 4] - LLM_gen_set(size_limit=-1)[source]¶
- Produce a list of upper cluster algebra elements corresponding to all vectors in \(\{0,1\}^n\). - INPUT: - B– a skew-symmetric matrix
- size_limit– a limit on how many vectors you want the function to return
 - OUTPUT: an array of elements in the upper cluster algebra - EXAMPLES: - sage: B = matrix([[0,1,0],[-1,0,1],[0,-1,0],[1,0,0],[0,1,0],[0,0,1]]) sage: C = ClusterSeed(B) sage: C.LLM_gen_set() [1, (x1 + x3)/x0, (x0*x4 + x2)/x1, (x0*x3*x4 + x1*x2 + x2*x3)/(x0*x1), (x1*x5 + 1)/x2, (x1^2*x5 + x1*x3*x5 + x1 + x3)/(x0*x2), (x0*x1*x4*x5 + x0*x4 + x2)/(x1*x2), (x0*x1*x3*x4*x5 + x0*x3*x4 + x1*x2 + x2*x3)/(x0*x1*x2)] - >>> from sage.all import * >>> B = matrix([[Integer(0),Integer(1),Integer(0)],[-Integer(1),Integer(0),Integer(1)],[Integer(0),-Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]]) >>> C = ClusterSeed(B) >>> C.LLM_gen_set() [1, (x1 + x3)/x0, (x0*x4 + x2)/x1, (x0*x3*x4 + x1*x2 + x2*x3)/(x0*x1), (x1*x5 + 1)/x2, (x1^2*x5 + x1*x3*x5 + x1 + x3)/(x0*x2), (x0*x1*x4*x5 + x0*x4 + x2)/(x1*x2), (x0*x1*x3*x4*x5 + x0*x3*x4 + x1*x2 + x2*x3)/(x0*x1*x2)] 
 - b_matrix()[source]¶
- Return the \(B\) -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] - >>> from sage.all import * >>> ClusterSeed(['A',Integer(4)]).b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] >>> ClusterSeed(['B',Integer(4)]).b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -2 0] >>> ClusterSeed(['D',Integer(4)]).b_matrix() [ 0 1 0 0] [-1 0 -1 -1] [ 0 1 0 0] [ 0 1 0 0] >>> ClusterSeed(QuiverMutationType([['A',Integer(2)],['B',Integer(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)[source]¶
- Return all \(B\)-matrices in the mutation class of - self.- INPUT: - depth– (default: infinity) integer or infinity, only seeds with distance at most- depthfrom- selfare returned
- up_to_equivalence– boolean (default:- True); if- True, only \(B\)-matrices up to equivalence are considered
 - EXAMPLES: - for examples see - b_matrix_class_iter()
 
 - b_matrix_class_iter(depth=+Infinity, up_to_equivalence=True)[source]¶
- Return an iterator through all \(B\)-matrices in the mutation class of - self.- INPUT: - depth– (default: infinity) integer or infinity, only seeds with distance at most- depthfrom- selfare returned
- up_to_equivalence– boolean (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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]) >>> it = S.b_matrix_class_iter() >>> 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] - >>> from sage.all import * >>> it = S.b_matrix_class_iter(depth=Integer(1)) >>> 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: b_matrix_class = list(it) sage: len(b_matrix_class) 14 sage: b_matrix_class[0] [ 0 1 0] [-1 0 -1] [ 0 1 0] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> it = S.b_matrix_class_iter(up_to_equivalence=False) >>> b_matrix_class = list(it) >>> len(b_matrix_class) 14 >>> b_matrix_class[Integer(0)] [ 0 1 0] [-1 0 -1] [ 0 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',[Integer(1),Integer(2)],Integer(1)]) >>> it = S.b_matrix_class_iter() >>> 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 - >>> from sage.all import * >>> S = ClusterSeed(['E',Integer(10)]) >>> it = S.b_matrix_class_iter(depth=Integer(3)) >>> len ([T for T in it]) 266 - For a cluster seed from an arbitrarily labelled digraph: - sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') sage: S = ClusterSeed(dg, frozen=['b']) sage: S.b_matrix_class() [ [ 0 0] [ 0 0] [0 0] [ 0 0] [ 0 0] [0 0] [-1 1], [-1 -1], [1 1] ] - >>> from sage.all import * >>> dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') >>> S = ClusterSeed(dg, frozen=['b']) >>> S.b_matrix_class() [ [ 0 0] [ 0 0] [0 0] [ 0 0] [ 0 0] [0 0] [-1 1], [-1 -1], [1 1] ] 
 - c_matrix(show_warnings=True)[source]¶
- Return all c-vectors of - self.- 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] sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False) sage: S.use_c_vectors(False); S.use_d_vectors(False); S.track_mutations(False) sage: S.c_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate c-vectors. Need to use c vectors. - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> S.c_matrix() [ 1 0 0] [ 0 0 -1] [ 0 -1 0] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False) >>> S.use_c_vectors(False); S.use_d_vectors(False); S.track_mutations(False) >>> S.c_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate c-vectors. Need to use c vectors. 
 - c_vector(k)[source]¶
- Return the - k-th c-vector of- self. It is obtained as the- k-th column vector of the bottom part of the- B-matrix of- self.- 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) (1, 0) sage: S = ClusterSeed(Matrix([[0,1],[-1,0],[1,0],[-1,1]])) sage: S.use_c_vectors(bot_is_c=True); S A seed for a cluster algebra of rank 2 with 2 frozen variables sage: S.c_vector(0) (1, -1) - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> [S.c_vector(k) for k in range(Integer(3))] [(1, 0, 0), (0, 0, -1), (0, -1, 0)] >>> S = ClusterSeed(Matrix([[Integer(0),Integer(1)],[-Integer(1),Integer(0)],[Integer(1),Integer(0)],[-Integer(1),Integer(1)]])); S A seed for a cluster algebra of rank 2 with 2 frozen variables >>> S.c_vector(Integer(0)) (1, 0) >>> S = ClusterSeed(Matrix([[Integer(0),Integer(1)],[-Integer(1),Integer(0)],[Integer(1),Integer(0)],[-Integer(1),Integer(1)]])) >>> S.use_c_vectors(bot_is_c=True); S A seed for a cluster algebra of rank 2 with 2 frozen variables >>> S.c_vector(Integer(0)) (1, -1) 
 - cluster()[source]¶
- Return a copy of 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.cluster() [x0, x1, x2] >>> S.mutate(Integer(1)) >>> S.cluster() [x0, (x0*x2 + 1)/x1, x2] >>> S.mutate(Integer(2)) >>> S.cluster() [x0, (x0*x2 + 1)/x1, (x0*x2 + x1 + 1)/(x1*x2)] >>> S.mutate([Integer(2),Integer(1)]) >>> S.cluster() [x0, x1, x2] 
 - cluster_class(depth=+Infinity, show_depth=False, up_to_equivalence=True)[source]¶
- Return the cluster class of - selfwith respect to certain constraints.- INPUT: - depth– (default: infinity) integer, only seeds with distance at most- depthfrom- selfare returned
- return_depth– (default:- False) if- True, ignored if- depthis set; returns the depth of the mutation class, i.e., the maximal distance from- selfof an element in the mutation class
- up_to_equivalence– (default:- True) if- True, only clusters up to equivalence are considered
 - EXAMPLES: - for examples see - cluster_class_iter()
 
 - cluster_class_iter(depth=+Infinity, show_depth=False, up_to_equivalence=True)[source]¶
- Return an iterator through all clusters in the mutation class of - self.- INPUT: - depth– (default: infinity) integer or infinity, only seeds with distance at most- depthfrom- selfare returned
- show_depth– boolean (default:- False); if- True, ignored if- depthis set; returns the depth of the mutation class, i.e., the maximal distance from- selfof an element in the mutation class
- up_to_equivalence– boolean (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: cluster_class = list(it) sage: len(cluster_class) 14 sage: cluster_class[0] [x0, x1, x2] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> it = S.cluster_class_iter() >>> cluster_class = list(it) >>> len(cluster_class) 14 >>> cluster_class[Integer(0)] [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] - >>> from sage.all import * >>> it = S.cluster_class_iter(depth=Integer(1)) >>> 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: _ = list(it) 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 - >>> from sage.all import * >>> it = S.cluster_class_iter(show_depth=True) >>> _ = list(it) 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 - 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: cluster_class = list(it) sage: len(cluster_class) 10 sage: cluster_class[0] [x0, x1] sage: cluster_class[-1] [x1, x0] - >>> from sage.all import * >>> it = S.cluster_class_iter(up_to_equivalence=False) >>> len([T for T in it]) 84 >>> it = ClusterSeed(['A',Integer(2)]).cluster_class_iter(up_to_equivalence=False) >>> cluster_class = list(it) >>> len(cluster_class) 10 >>> cluster_class[Integer(0)] [x0, x1] >>> cluster_class[-Integer(1)] [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)] - >>> from sage.all import * >>> S = ClusterSeed(['A',[Integer(1),Integer(1)],Integer(1)]) >>> it = S.cluster_class_iter() >>> next(it) [x0, x1] >>> next(it) [x0, (x0^2 + 1)/x1] >>> next(it) [(x1^2 + 1)/x0, x1] >>> next(it) [(x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2), (x0^2 + 1)/x1] >>> next(it) [(x1^2 + 1)/x0, (x1^4 + x0^2 + 2*x1^2 + 1)/(x0^2*x1)] >>> it = S.cluster_class_iter(depth=Integer(3)) >>> 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)] - For a cluster seed from an arbitrarily labelled digraph: - sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') sage: S = ClusterSeed(dg, frozen=['b']) sage: S.cluster_class() [[a, c], [a, (b + 1)/c], [(b + 1)/a, c], [(b + 1)/a, (b + 1)/c]] sage: S2 = ClusterSeed(dg, frozen=[]) sage: S2.cluster_class()[0] [a, b, c] - >>> from sage.all import * >>> dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') >>> S = ClusterSeed(dg, frozen=['b']) >>> S.cluster_class() [[a, c], [a, (b + 1)/c], [(b + 1)/a, c], [(b + 1)/a, (b + 1)/c]] >>> S2 = ClusterSeed(dg, frozen=[]) >>> S2.cluster_class()[Integer(0)] [a, b, c] 
 - cluster_index(cluster_str)[source]¶
- Return the index of a cluster if - use_fpolysis on.- INPUT: - cluster_str– the string to look for in the cluster
 - OUTPUT: integer or - Noneif the string is not a cluster variable- EXAMPLES: - sage: S = ClusterSeed(['A', 4], user_labels=['x', 'y', 'z', 'w']); S.mutate('x') sage: S.cluster_index('x') sage: S.cluster_index('(y+1)/x') 0 - >>> from sage.all import * >>> S = ClusterSeed(['A', Integer(4)], user_labels=['x', 'y', 'z', 'w']); S.mutate('x') >>> S.cluster_index('x') >>> S.cluster_index('(y+1)/x') 0 
 - cluster_variable(k)[source]¶
- Generates a cluster variable using F-polynomials. - EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: S.mutate([0,1]) sage: S.cluster_variable(0) (x1 + 1)/x0 sage: S.cluster_variable(1) (x0*x2 + x1 + 1)/(x0*x1) - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.mutate([Integer(0),Integer(1)]) >>> S.cluster_variable(Integer(0)) (x1 + 1)/x0 >>> S.cluster_variable(Integer(1)) (x0*x2 + x1 + 1)/(x0*x1) 
 - coefficient(k)[source]¶
- Return the coefficient of - selfat index- k, or vertex- kif- kis not an index.- 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> [S.coefficient(k) for k in range(Integer(3))] [y0, 1/y2, 1/y1] 
 - coefficients()[source]¶
- Return 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> S.coefficients() [y0, 1/y2, 1/y1] 
 - d_matrix(show_warnings=True)[source]¶
- Return the matrix of d-vectors of - self.- EXAMPLES: - sage: S = ClusterSeed(['A',4]); S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S.mutate([1,2,1,0,1,3]); S.d_matrix() [1 1 0 1] [1 1 1 1] [1 0 1 1] [0 0 0 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]); S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S.mutate([Integer(1),Integer(2),Integer(1),Integer(0),Integer(1),Integer(3)]); S.d_matrix() [1 1 0 1] [1 1 1 1] [1 0 1 1] [0 0 0 1] 
 - d_vector(k)[source]¶
- Return the - k-th d-vector of- self. This is the exponent vector of the denominator of the- k-th cluster variable.- EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: S.mutate([2,1,2]) sage: [S.d_vector(k) for k in range(3)] [(-1, 0, 0), (0, 1, 1), (0, 1, 0)] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> [S.d_vector(k) for k in range(Integer(3))] [(-1, 0, 0), (0, 1, 1), (0, 1, 0)] 
 - exchangeable_part()[source]¶
- Return 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(sort=True), frozen=[3]) sage: T.quiver().digraph().edges(sort=True) [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))] sage: T.exchangeable_part().quiver().digraph().edges(sort=True) [(0, 1, (1, -1)), (2, 1, (1, -1))] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]) >>> T = ClusterSeed(S.quiver().digraph().edges(sort=True), frozen=[Integer(3)]) >>> T.quiver().digraph().edges(sort=True) [(0, 1, (1, -1)), (2, 1, (1, -1)), (2, 3, (1, -1))] >>> T.exchangeable_part().quiver().digraph().edges(sort=True) [(0, 1, (1, -1)), (2, 1, (1, -1))] 
 - f_polynomial(k)[source]¶
- Return the - k-th F-polynomial of- self. It is obtained from the- k-th cluster variable by setting all \(x_i\) to \(1\).- 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]])) sage: S.use_c_vectors(bot_is_c=True); 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() [y0 + y1, 1] sage: T.f_polynomials() [y0 + 1, 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> [S.f_polynomial(k) for k in range(Integer(3))] [1, y1*y2 + y2 + 1, y1 + 1] >>> S = ClusterSeed(Matrix([[Integer(0),Integer(1)],[-Integer(1),Integer(0)],[Integer(1),Integer(0)],[-Integer(1),Integer(1)]])) >>> S.use_c_vectors(bot_is_c=True); S A seed for a cluster algebra of rank 2 with 2 frozen variables >>> T = ClusterSeed(Matrix([[Integer(0),Integer(1)],[-Integer(1),Integer(0)]])).principal_extension(); T A seed for a cluster algebra of rank 2 with principal coefficients >>> S.mutate(Integer(0)) >>> T.mutate(Integer(0)) >>> S.f_polynomials() [y0 + y1, 1] >>> T.f_polynomials() [y0 + 1, 1] 
 - f_polynomials()[source]¶
- Return all F-polynomials of - self. These are obtained from the cluster variables by setting all \(x_i\)’s to \(1\).- 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> S.f_polynomials() [1, y1*y2 + y2 + 1, y1 + 1] 
 - find_upper_bound(verbose=False)[source]¶
- Return the upper bound of the given cluster algebra as a quotient ring. - The upper bound is the intersection of the Laurent polynomial rings of the initial cluster and its neighboring clusters. As such, it always contains both the cluster algebra and the upper cluster algebra. This function uses the algorithm from [MM2015]. - When the initial seed is totally coprime (for example, when the unfrozen part of the exchange matrix has full rank), the upper bound is equal to the upper cluster algebra by [BFZ2005]. - Warning - The computation time grows rapidly with the size of the seed and the number of steps. For most seeds larger than four vertices, the algorithm may take an infeasible amount of time. Additionally, it will run forever without terminating whenever the upper bound is infinitely-generated (such as the example in [Spe2013]). - INPUT: - verbose– boolean (default:- False); if- True, prints output during the computation
 - EXAMPLES: - finite type: - sage: S = ClusterSeed(['A',3]) sage: S.find_upper_bound() Quotient of Multivariate Polynomial Ring in x0, x1, x2, x0p, x1p, x2p, z0 over Rational Field by the ideal (x0*x0p - x1 - 1, x1*x1p - x0*x2 - 1, x2*x2p - x1 - 1, x0*z0 - x2p, x1*z0 + z0 - x0p*x2p, x2*z0 - x0p, x1p*z0 + z0 - x0p*x1p*x2p + x1 + 1) - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.find_upper_bound() Quotient of Multivariate Polynomial Ring in x0, x1, x2, x0p, x1p, x2p, z0 over Rational Field by the ideal (x0*x0p - x1 - 1, x1*x1p - x0*x2 - 1, x2*x2p - x1 - 1, x0*z0 - x2p, x1*z0 + z0 - x0p*x2p, x2*z0 - x0p, x1p*z0 + z0 - x0p*x1p*x2p + x1 + 1) 
- Markov: - sage: B = matrix([[0,2,-2],[-2,0,2],[2,-2,0]]) sage: S = ClusterSeed(B) sage: S.find_upper_bound() Quotient of Multivariate Polynomial Ring in x0, x1, x2, x0p, x1p, x2p, z0 over Rational Field by the ideal (x0*x0p - x2^2 - x1^2, x1*x1p - x2^2 - x0^2, x2*x2p - x1^2 - x0^2, x0p*x1p*x2p - x0*x1*x2p - x0*x2*x1p - x1*x2*x0p - 2*x0*x1*x2, x0^3*z0 - x1p*x2p + x1*x2, x0*x1*z0 - x2p - x2, x1^3*z0 - x0p*x2p + x0*x2, x0*x2*z0 - x1p - x1, x1*x2*z0 - x0p - x0, x2^3*z0 - x0p*x1p + x0*x1) - >>> from sage.all import * >>> B = matrix([[Integer(0),Integer(2),-Integer(2)],[-Integer(2),Integer(0),Integer(2)],[Integer(2),-Integer(2),Integer(0)]]) >>> S = ClusterSeed(B) >>> S.find_upper_bound() Quotient of Multivariate Polynomial Ring in x0, x1, x2, x0p, x1p, x2p, z0 over Rational Field by the ideal (x0*x0p - x2^2 - x1^2, x1*x1p - x2^2 - x0^2, x2*x2p - x1^2 - x0^2, x0p*x1p*x2p - x0*x1*x2p - x0*x2*x1p - x1*x2*x0p - 2*x0*x1*x2, x0^3*z0 - x1p*x2p + x1*x2, x0*x1*z0 - x2p - x2, x1^3*z0 - x0p*x2p + x0*x2, x0*x2*z0 - x1p - x1, x1*x2*z0 - x0p - x0, x2^3*z0 - x0p*x1p + x0*x1) 
 
 - first_green_vertex()[source]¶
- Return the first green vertex of - self.- A vertex is defined to be green if its c-vector has all nonpositive entries. More information on green vertices can be found at [BDP2013] - EXAMPLES: - sage: ClusterSeed(['A',3]).principal_extension().first_green_vertex() 0 sage: ClusterSeed(['A',[3,3],1]).principal_extension().first_green_vertex() 0 - >>> from sage.all import * >>> ClusterSeed(['A',Integer(3)]).principal_extension().first_green_vertex() 0 >>> ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension().first_green_vertex() 0 
 - first_red_vertex()[source]¶
- Return the first red vertex of - self.- A vertex is defined to be red if its c-vector has all nonnegative entries. More information on red vertices can be found at [BDP2013]. - EXAMPLES: - sage: ClusterSeed(['A',3]).principal_extension().first_red_vertex() sage: ClusterSeed(['A',[3,3],1]).principal_extension().first_red_vertex() sage: Q = ClusterSeed(['A',[3,3],1]).principal_extension() sage: Q.mutate(1) sage: Q.first_red_vertex() 1 - >>> from sage.all import * >>> ClusterSeed(['A',Integer(3)]).principal_extension().first_red_vertex() >>> ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension().first_red_vertex() >>> Q = ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension() >>> Q.mutate(Integer(1)) >>> Q.first_red_vertex() 1 
 - first_urban_renewal()[source]¶
- Return the first urban renewal vertex. - An urban renewal vertex is one in which there are two arrows pointing toward the vertex and two arrows pointing away. - EXAMPLES: - sage: G = ClusterSeed(['GR',[4,9]]); G.first_urban_renewal() 5 - >>> from sage.all import * >>> G = ClusterSeed(['GR',[Integer(4),Integer(9)]]); G.first_urban_renewal() 5 
 - free_vertices()[source]¶
- Return the list of exchangeable vertices of - self.- EXAMPLES: - sage: S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ....: frozen=['b', 'd']) sage: S.free_vertices() ['a', 'c', 'e'] sage: S = ClusterSeed(DiGraph([[5, 'b']])) sage: S.free_vertices() [5, 'b'] - >>> from sage.all import * >>> S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ... frozen=['b', 'd']) >>> S.free_vertices() ['a', 'c', 'e'] >>> S = ClusterSeed(DiGraph([[Integer(5), 'b']])) >>> S.free_vertices() [5, 'b'] 
 - frozen_vertices()[source]¶
- Return the list of frozen vertices of - self.- EXAMPLES: - sage: S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ....: frozen=['b', 'd']) sage: sorted(S.frozen_vertices()) ['b', 'd'] - >>> from sage.all import * >>> S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['c', 'd'], ['e', 'd']]), ... frozen=['b', 'd']) >>> sorted(S.frozen_vertices()) ['b', 'd'] 
 - g_matrix(show_warnings=True)[source]¶
- Return the matrix of all g-vectors of - self. These are the degree vectors of the cluster variables after setting all \(y_i\)’s to \(0\).- 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: S.mutate([0,1]) sage: S.g_matrix() [-1 -1 0] [ 1 0 0] [ 0 0 1] sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False); S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_c_vectors(False); S.use_fpolys(False) sage: S.track_mutations(False); S.g_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate g-vectors. Need to use g vectors. - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> S.g_matrix() [ 1 0 0] [ 0 0 -1] [ 0 -1 0] >>> S = ClusterSeed(['A',Integer(3)]) >>> S.mutate([Integer(0),Integer(1)]) >>> S.g_matrix() [-1 -1 0] [ 1 0 0] [ 0 0 1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False); S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_c_vectors(False); S.use_fpolys(False) >>> S.track_mutations(False); S.g_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate g-vectors. Need to use g vectors. 
 - g_vector(k)[source]¶
- Return the - k-th g-vector of- self. This is the degree vector of the- k-th cluster variable after setting all \(y_i\)’s to \(0\).- 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)] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutate([Integer(2),Integer(1),Integer(2)]) >>> [S.g_vector(k) for k in range(Integer(3))] [(1, 0, 0), (0, 0, -1), (0, -1, 0)] 
 - get_upper_cluster_algebra_element(a)[source]¶
- Compute an element in the upper cluster algebra of \(B\) corresponding to the vector \(a \in \ZZ^n\). - See [LLM2014] for more details. - INPUT: - B– a skew-symmetric matrix. Must have the same number of columns as the length of the vectors in \(vd\)
- a– a vector in \(\ZZ^n\) where \(n\) is the number of columns in \(B\)
 - OUTPUT: an element in the upper cluster algebra. Depending on the input it may or may not be irreducible - EXAMPLES: - sage: B = matrix([[0,3,-3],[-3,0,3],[3,-3,0],[1,0,0],[0,1,0],[0,0,1]]) sage: C = ClusterSeed(B) sage: C.get_upper_cluster_algebra_element([1,1,0]) (x0^3*x2^3*x3*x4 + x2^6*x3 + x1^3*x2^3)/(x0*x1) sage: C.get_upper_cluster_algebra_element([1,1,1]) x0^2*x1^2*x2^2*x3*x4*x5 + x0^2*x1^2*x2^2 sage: B = matrix([[0,3,0],[-3,0,3],[0,-3,0]]) sage: C = ClusterSeed(B) sage: C.get_upper_cluster_algebra_element([1,1,0]) (x1^3*x2^3 + x0^3 + x2^3)/(x0*x1) sage: C.get_upper_cluster_algebra_element([1,1,1]) (x0^3*x1^3 + x1^3*x2^3 + x0^3 + x2^3)/(x0*x1*x2) sage: B = matrix([[0,2],[-3,0],[4,-5]]) sage: C = ClusterSeed(B) sage: C.get_upper_cluster_algebra_element([1,1]) (x2^9 + x1^3*x2^5 + x0^2*x2^4)/(x0*x1) sage: B = matrix([[0,3,-5],[-3,0,4],[5,-4,0]]) sage: C = ClusterSeed(B) sage: C.get_upper_cluster_algebra_element([1,1,1]) x0^4*x1^2*x2^3 + x0^2*x1^3*x2^4 - >>> from sage.all import * >>> B = matrix([[Integer(0),Integer(3),-Integer(3)],[-Integer(3),Integer(0),Integer(3)],[Integer(3),-Integer(3),Integer(0)],[Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1)]]) >>> C = ClusterSeed(B) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1),Integer(0)]) (x0^3*x2^3*x3*x4 + x2^6*x3 + x1^3*x2^3)/(x0*x1) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1),Integer(1)]) x0^2*x1^2*x2^2*x3*x4*x5 + x0^2*x1^2*x2^2 >>> B = matrix([[Integer(0),Integer(3),Integer(0)],[-Integer(3),Integer(0),Integer(3)],[Integer(0),-Integer(3),Integer(0)]]) >>> C = ClusterSeed(B) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1),Integer(0)]) (x1^3*x2^3 + x0^3 + x2^3)/(x0*x1) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1),Integer(1)]) (x0^3*x1^3 + x1^3*x2^3 + x0^3 + x2^3)/(x0*x1*x2) >>> B = matrix([[Integer(0),Integer(2)],[-Integer(3),Integer(0)],[Integer(4),-Integer(5)]]) >>> C = ClusterSeed(B) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1)]) (x2^9 + x1^3*x2^5 + x0^2*x2^4)/(x0*x1) >>> B = matrix([[Integer(0),Integer(3),-Integer(5)],[-Integer(3),Integer(0),Integer(4)],[Integer(5),-Integer(4),Integer(0)]]) >>> C = ClusterSeed(B) >>> C.get_upper_cluster_algebra_element([Integer(1),Integer(1),Integer(1)]) x0^4*x1^2*x2^3 + x0^2*x1^3*x2^4 
 - greedy(a1, a2, algorithm='by_recursion')[source]¶
- Return the greedy element \(x[a_1,a_2]\) assuming that - selfis 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 \(x_1\) and \(x_2\) to be one.- See [LLZ2014] 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) - >>> from sage.all import * >>> S = ClusterSeed(['R2', [Integer(3), Integer(3)]]) >>> S.greedy(Integer(4), Integer(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) >>> S.greedy(Integer(4), Integer(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) >>> S.greedy(Integer(4), Integer(4), 'just_numbers') 35 >>> S = ClusterSeed(['R2', [Integer(2), Integer(2)]]) >>> S.greedy(Integer(1), Integer(2)) (x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2) >>> S.greedy(Integer(1), Integer(2), 'by_combinatorics') (x0^4 + 2*x0^2 + x1^2 + 1)/(x0*x1^2) 
 - green_vertices()[source]¶
- Return the list of green vertices of - self.- A vertex is defined to be green if its c-vector has all nonpositive entries. More information on green vertices can be found at [BDP2013] - OUTPUT: the green vertices as a list of integers - EXAMPLES: - sage: ClusterSeed(['A',3]).principal_extension().green_vertices() [0, 1, 2] sage: ClusterSeed(['A',[3,3],1]).principal_extension().green_vertices() [0, 1, 2, 3, 4, 5] - >>> from sage.all import * >>> ClusterSeed(['A',Integer(3)]).principal_extension().green_vertices() [0, 1, 2] >>> ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension().green_vertices() [0, 1, 2, 3, 4, 5] 
 - ground_field()[source]¶
- Return the ground field of the cluster of - self.- EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: S.ground_field() Multivariate Polynomial Ring in x0, x1, x2, y0, y1, y2 over Rational Field - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.ground_field() Multivariate Polynomial Ring in x0, x1, x2, y0, y1, y2 over Rational Field 
 - highest_degree_denominator(filter=None)[source]¶
- Return the vertex of the cluster polynomial with highest degree in the denominator. - INPUT: - filter– list or iterable
 - OUTPUT: integer - EXAMPLES: - sage: B = matrix([[0,-1,0,-1,1,1], [1,0,1,0,-1,-1], [0,-1,0,-1,1,1], ....: [1,0,1,0,-1,-1], [-1,1,-1,1,0,0], [-1,1,-1,1,0,0]]) sage: C = ClusterSeed(B).principal_extension(); C.mutate([0,1,2,4,3,2,5,4,3]) sage: C.highest_degree_denominator() 5 - >>> from sage.all import * >>> B = matrix([[Integer(0),-Integer(1),Integer(0),-Integer(1),Integer(1),Integer(1)], [Integer(1),Integer(0),Integer(1),Integer(0),-Integer(1),-Integer(1)], [Integer(0),-Integer(1),Integer(0),-Integer(1),Integer(1),Integer(1)], ... [Integer(1),Integer(0),Integer(1),Integer(0),-Integer(1),-Integer(1)], [-Integer(1),Integer(1),-Integer(1),Integer(1),Integer(0),Integer(0)], [-Integer(1),Integer(1),-Integer(1),Integer(1),Integer(0),Integer(0)]]) >>> C = ClusterSeed(B).principal_extension(); C.mutate([Integer(0),Integer(1),Integer(2),Integer(4),Integer(3),Integer(2),Integer(5),Integer(4),Integer(3)]) >>> C.highest_degree_denominator() 5 
 - interact(fig_size=1, circular=True)[source]¶
- Start an interactive window for cluster seed mutations. - Only in Jupyter notebook mode. - INPUT: - fig_size– (default: 1) factor by which the size of the plot is multiplied
- circular– boolean (default:- True); if- True, the circular plot is chosen, otherwise >>spring<< is used
 
 - is_acyclic()[source]¶
- Return - Trueiff- selfis 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 - >>> from sage.all import * >>> ClusterSeed(['A',Integer(4)]).is_acyclic() True >>> ClusterSeed(['A',[Integer(2),Integer(1)],Integer(1)]).is_acyclic() True >>> ClusterSeed([[Integer(0),Integer(1)],[Integer(1),Integer(2)],[Integer(2),Integer(0)]]).is_acyclic() False 
 - is_bipartite(return_bipartition=False)[source]¶
- Return - Trueiff- selfis bipartite (i.e., if the underlying quiver is bipartite).- INPUT: - return_bipartition– boolean (default:- False); if- True, the bipartition is returned in the case of- selfbeing bipartite
 - EXAMPLES: - sage: ClusterSeed(['A',[3,3],1]).is_bipartite() True sage: ClusterSeed(['A',[4,3],1]).is_bipartite() False - >>> from sage.all import * >>> ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).is_bipartite() True >>> ClusterSeed(['A',[Integer(4),Integer(3)],Integer(1)]).is_bipartite() False 
 - is_finite()[source]¶
- Return - Trueif- selfis 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 - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.is_finite() True >>> S = ClusterSeed(['A',[Integer(2),Integer(2)],Integer(1)]) >>> S.is_finite() False 
 - is_mutation_finite(nr_of_checks=None, return_path=False)[source]¶
- Return - Trueif- selfis of finite mutation type.- INPUT: - nr_of_checks– (default:- None) number of mutations applied; standard is 500 times the number of vertices of- self
- return_path– boolean (default:- False); if- True, in case of- selfnot being mutation finite, a path from- selfto 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 \(b_{ij}*b_{ji} > -4\). 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 - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(10)]) >>> S._mutation_type = None >>> S.is_mutation_finite() True >>> S = ClusterSeed([(Integer(0),Integer(1)),(Integer(1),Integer(2)),(Integer(2),Integer(3)),(Integer(3),Integer(4)),(Integer(4),Integer(5)),(Integer(5),Integer(6)),(Integer(6),Integer(7)),(Integer(7),Integer(8)),(Integer(2),Integer(9))]) >>> S.is_mutation_finite() False 
 - m()[source]¶
- Return 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 - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.n() 3 >>> S.m() 0 >>> S = S.principal_extension() >>> S.m() 3 
 - most_decreased_denominator_after_mutation()[source]¶
- Return the vertex that will produce the most decrease in denominator degrees after mutation. - EXAMPLES: - sage: S = ClusterSeed(['A',5]) sage: S.mutate([0,2,3,1,2,3,1,2,0,2,3]) sage: S.most_decreased_denominator_after_mutation() 2 - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(5)]) >>> S.mutate([Integer(0),Integer(2),Integer(3),Integer(1),Integer(2),Integer(3),Integer(1),Integer(2),Integer(0),Integer(2),Integer(3)]) >>> S.most_decreased_denominator_after_mutation() 2 
 - most_decreased_edge_after_mutation()[source]¶
- Return the vertex that will produce the least degrees after mutation. - EXAMPLES: - sage: S = ClusterSeed(['A',5]) sage: S.mutate([0,2,3,1,2,3,1,2,0,2,3]) sage: S.most_decreased_edge_after_mutation() 2 - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(5)]) >>> S.mutate([Integer(0),Integer(2),Integer(3),Integer(1),Integer(2),Integer(3),Integer(1),Integer(2),Integer(0),Integer(2),Integer(3)]) >>> S.most_decreased_edge_after_mutation() 2 
 - mutate(sequence, inplace=True, input_type=None)[source]¶
- Mutate - selfat a vertex or a sequence of vertices.- INPUT: - sequence– a vertex of- self, an iterator of vertices of- self, a function which takes in the- ClusterSeedand returns a vertex or an iterator of vertices, or a string representing a type of vertices to mutate
- inplace– boolean (default:- True); if- False, the result is returned, otherwise- selfis modified
- input_type– (default:- None) indicates the type of data contained in the sequence
 - Possible values for vertex types in - sequenceare:- 'first_source'– mutates at first found source vertex
- 'sources'– mutates at all sources
- 'first_sink'– mutates at first sink
- 'sinks'– mutates at all sink vertices
- 'green'– mutates at the first green vertex
- 'red'– mutates at the first red vertex
- 'urban_renewal'or- 'urban'– mutates at first urban renewal vertex
- 'all_urban_renewals'or- 'all_urban'– mutates at all urban renewal vertices
 - For - input_type, if no value is given, preference will be given to vertex names, then indices, then cluster variables. If all input is not of the same type, an error is given. Possible values for- input_typeare:- 'vertices'– interprets the input sequence as vertices
- 'indices'– interprets the input sequence as indices
- 'cluster_vars'– interprets the input sequence as cluster variables. This must be selected if inputting a sequence of cluster variables.
 - 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 sage: Q = ClusterSeed(['A',3]);Q.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] sage: Q.mutate('first_sink');Q.b_matrix() [ 0 -1 0] [ 1 0 1] [ 0 -1 0] sage: def last_vertex(self): return self._n - 1 sage: Q.mutate(last_vertex); Q.b_matrix() [ 0 -1 0] [ 1 0 -1] [ 0 1 0] sage: S = ClusterSeed(['A', 4], user_labels=['a', 'b', 'c', 'd']) sage: S.mutate('a'); S.mutate('(b+1)/a') sage: S.cluster() [a, b, c, d] sage: S = ClusterSeed(['A', 4], user_labels=['a', 'b', 'c']) Traceback (most recent call last): ... ValueError: the number of user-defined labels is not the number of exchangeable and frozen variables sage: S = ClusterSeed(['A', 4], user_labels=['x', 'y', 'w', 'z']) sage: S.mutate('x') sage: S.cluster() [(y + 1)/x, y, w, z] sage: S.mutate('(y+1)/x') sage: S.cluster() [x, y, w, z] sage: S.mutate('y') sage: S.cluster() [x, (x*w + 1)/y, w, z] sage: S.mutate('(x*w+1)/y') sage: S.cluster() [x, y, w, z] sage: S = ClusterSeed(['A', 4], user_labels=[[1, 2], [2, 3], [4, 5], [5, 6]]) sage: S.cluster() [x_1_2, x_2_3, x_4_5, x_5_6] sage: S.mutate('[1,2]') sage: S.cluster() [(x_2_3 + 1)/x_1_2, x_2_3, x_4_5, x_5_6] sage: S = ClusterSeed(['A', 4], user_labels=[[1, 2], [2, 3], [4, 5], [5, 6]], ....: user_labels_prefix='P'); sage: S.cluster() [P_1_2, P_2_3, P_4_5, P_5_6] sage: S.mutate('[1,2]') sage: S.cluster() [(P_2_3 + 1)/P_1_2, P_2_3, P_4_5, P_5_6] sage: S.mutate('P_4_5') sage: S.cluster() [(P_2_3 + 1)/P_1_2, P_2_3, (P_2_3*P_5_6 + 1)/P_4_5, P_5_6] sage: S = ClusterSeed(['A', 4]) sage: S.mutate([0, 1, 0, 1, 0, 2, 1]) sage: T = ClusterSeed(S) sage: S.use_fpolys(False) sage: S.use_g_vectors(False) sage: S.use_c_vectors(False) sage: S._C sage: S._G sage: S._F sage: S.g_matrix() [ 0 -1 0 0] [ 1 1 1 0] [ 0 0 -1 0] [ 0 0 1 1] sage: S.c_matrix() [ 1 -1 0 0] [ 1 0 0 0] [ 1 0 -1 1] [ 0 0 0 1] sage: S.f_polynomials() == T.f_polynomials() True sage: S.cluster() == T.cluster() True sage: S._mut_path [0, 1, 0, 1, 0, 2, 1] sage: S = ClusterSeed(DiGraph([[1, 2], [2, 'c']])) sage: S.mutate(1) Input can be ambiguously interpreted as both vertices and indices. Mutating at vertices by default. sage: S.cluster() [(x2 + 1)/x1, x2, c] sage: S.mutate(1, input_type='indices') sage: S.cluster() [(x2 + 1)/x1, (x2*c + x1 + c)/(x1*x2), c] sage: S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['d', 'b']])) sage: S.mutate(['a', 'b', 'a', 'b', 'a']) sage: S.cluster() [b, a, c, d] sage: S.mutate('a') Input can be ambiguously interpreted as both vertices and cluster variables. Mutating at vertices by default. sage: S.cluster() [(a*c*d + 1)/b, a, c, d] sage: S.mutate('a', input_type='cluster_vars') sage: S.cluster() [(a*c*d + 1)/b, (a*c*d + b + 1)/(a*b), c, d] sage: S.mutate(['(a*c*d + 1)/b', 'd']) sage: S.cluster() [(b + 1)/a, (a*c*d + b + 1)/(a*b), c, (a*c*d + b^2 + 2*b + 1)/(a*b*d)] sage: S = ClusterSeed(DiGraph([[5, 'b']])) sage: S.mutate(5) sage: S.cluster() [(b + 1)/x5, b] sage: S.mutate([5]) sage: S.cluster() [x5, b] sage: S.mutate(0) sage: S.cluster() [(b + 1)/x5, b] sage: S = ClusterSeed(DiGraph([[1, 2]])) sage: S.cluster() [x1, x2] sage: S.mutate(1) Input can be ambiguously interpreted as both vertices and indices. Mutating at vertices by default. sage: S.cluster() [(x2 + 1)/x1, x2] sage: S = ClusterSeed(DiGraph([[-1, 0], [0, 1]])) sage: S.cluster() [xneg1, x0, x1] sage: S.mutate(-1);S.cluster() [(x0 + 1)/xneg1, x0, x1] sage: S.mutate(0, input_type='vertices');S.cluster() [(x0 + 1)/xneg1, (x0*x1 + xneg1 + x1)/(xneg1*x0), x1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]); S.b_matrix() [ 0 1 0 0] [-1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] >>> S.mutate(Integer(0)); S.b_matrix() [ 0 -1 0 0] [ 1 0 -1 0] [ 0 1 0 1] [ 0 0 -1 0] >>> T = S.mutate(Integer(0), inplace=False); T A seed for a cluster algebra of rank 4 of type ['A', 4] >>> S.mutate(Integer(0)) >>> S == T True >>> S.mutate([Integer(0),Integer(1),Integer(0)]) >>> S.b_matrix() [ 0 -1 1 0] [ 1 0 0 0] [-1 0 0 1] [ 0 0 -1 0] >>> S = ClusterSeed(QuiverMutationType([['A',Integer(1)],['A',Integer(3)]])) >>> S.b_matrix() [ 0 0 0 0] [ 0 0 1 0] [ 0 -1 0 -1] [ 0 0 1 0] >>> T = S.mutate(Integer(0),inplace=False) >>> S == T False >>> Q = ClusterSeed(['A',Integer(3)]);Q.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] >>> Q.mutate('first_sink');Q.b_matrix() [ 0 -1 0] [ 1 0 1] [ 0 -1 0] >>> def last_vertex(self): return self._n - Integer(1) >>> Q.mutate(last_vertex); Q.b_matrix() [ 0 -1 0] [ 1 0 -1] [ 0 1 0] >>> S = ClusterSeed(['A', Integer(4)], user_labels=['a', 'b', 'c', 'd']) >>> S.mutate('a'); S.mutate('(b+1)/a') >>> S.cluster() [a, b, c, d] >>> S = ClusterSeed(['A', Integer(4)], user_labels=['a', 'b', 'c']) Traceback (most recent call last): ... ValueError: the number of user-defined labels is not the number of exchangeable and frozen variables >>> S = ClusterSeed(['A', Integer(4)], user_labels=['x', 'y', 'w', 'z']) >>> S.mutate('x') >>> S.cluster() [(y + 1)/x, y, w, z] >>> S.mutate('(y+1)/x') >>> S.cluster() [x, y, w, z] >>> S.mutate('y') >>> S.cluster() [x, (x*w + 1)/y, w, z] >>> S.mutate('(x*w+1)/y') >>> S.cluster() [x, y, w, z] >>> S = ClusterSeed(['A', Integer(4)], user_labels=[[Integer(1), Integer(2)], [Integer(2), Integer(3)], [Integer(4), Integer(5)], [Integer(5), Integer(6)]]) >>> S.cluster() [x_1_2, x_2_3, x_4_5, x_5_6] >>> S.mutate('[1,2]') >>> S.cluster() [(x_2_3 + 1)/x_1_2, x_2_3, x_4_5, x_5_6] >>> S = ClusterSeed(['A', Integer(4)], user_labels=[[Integer(1), Integer(2)], [Integer(2), Integer(3)], [Integer(4), Integer(5)], [Integer(5), Integer(6)]], ... user_labels_prefix='P'); >>> S.cluster() [P_1_2, P_2_3, P_4_5, P_5_6] >>> S.mutate('[1,2]') >>> S.cluster() [(P_2_3 + 1)/P_1_2, P_2_3, P_4_5, P_5_6] >>> S.mutate('P_4_5') >>> S.cluster() [(P_2_3 + 1)/P_1_2, P_2_3, (P_2_3*P_5_6 + 1)/P_4_5, P_5_6] >>> S = ClusterSeed(['A', Integer(4)]) >>> S.mutate([Integer(0), Integer(1), Integer(0), Integer(1), Integer(0), Integer(2), Integer(1)]) >>> T = ClusterSeed(S) >>> S.use_fpolys(False) >>> S.use_g_vectors(False) >>> S.use_c_vectors(False) >>> S._C >>> S._G >>> S._F >>> S.g_matrix() [ 0 -1 0 0] [ 1 1 1 0] [ 0 0 -1 0] [ 0 0 1 1] >>> S.c_matrix() [ 1 -1 0 0] [ 1 0 0 0] [ 1 0 -1 1] [ 0 0 0 1] >>> S.f_polynomials() == T.f_polynomials() True >>> S.cluster() == T.cluster() True >>> S._mut_path [0, 1, 0, 1, 0, 2, 1] >>> S = ClusterSeed(DiGraph([[Integer(1), Integer(2)], [Integer(2), 'c']])) >>> S.mutate(Integer(1)) Input can be ambiguously interpreted as both vertices and indices. Mutating at vertices by default. >>> S.cluster() [(x2 + 1)/x1, x2, c] >>> S.mutate(Integer(1), input_type='indices') >>> S.cluster() [(x2 + 1)/x1, (x2*c + x1 + c)/(x1*x2), c] >>> S = ClusterSeed(DiGraph([['a', 'b'], ['c', 'b'], ['d', 'b']])) >>> S.mutate(['a', 'b', 'a', 'b', 'a']) >>> S.cluster() [b, a, c, d] >>> S.mutate('a') Input can be ambiguously interpreted as both vertices and cluster variables. Mutating at vertices by default. >>> S.cluster() [(a*c*d + 1)/b, a, c, d] >>> S.mutate('a', input_type='cluster_vars') >>> S.cluster() [(a*c*d + 1)/b, (a*c*d + b + 1)/(a*b), c, d] >>> S.mutate(['(a*c*d + 1)/b', 'd']) >>> S.cluster() [(b + 1)/a, (a*c*d + b + 1)/(a*b), c, (a*c*d + b^2 + 2*b + 1)/(a*b*d)] >>> S = ClusterSeed(DiGraph([[Integer(5), 'b']])) >>> S.mutate(Integer(5)) >>> S.cluster() [(b + 1)/x5, b] >>> S.mutate([Integer(5)]) >>> S.cluster() [x5, b] >>> S.mutate(Integer(0)) >>> S.cluster() [(b + 1)/x5, b] >>> S = ClusterSeed(DiGraph([[Integer(1), Integer(2)]])) >>> S.cluster() [x1, x2] >>> S.mutate(Integer(1)) Input can be ambiguously interpreted as both vertices and indices. Mutating at vertices by default. >>> S.cluster() [(x2 + 1)/x1, x2] >>> S = ClusterSeed(DiGraph([[-Integer(1), Integer(0)], [Integer(0), Integer(1)]])) >>> S.cluster() [xneg1, x0, x1] >>> S.mutate(-Integer(1));S.cluster() [(x0 + 1)/xneg1, x0, x1] >>> S.mutate(Integer(0), input_type='vertices');S.cluster() [(x0 + 1)/xneg1, (x0*x1 + xneg1 + x1)/(xneg1*x0), x1] 
 - mutation_analysis(options=['all'], filter=None)[source]¶
- Run an analysis of all potential mutation options. Note that this might take a long time on large seeds. - Note - Edges are only returned if we have a non-valued quiver. Green and red vertices are only returned if the cluster is principal. - INPUT: - options– (default:- ['all']) a list of mutation options
- filter– (default:- None) a vertex or interval of vertices to limit our search to
 - Possible options are: - 'all'– all options below
- 'edges'– number of edges (works with skew-symmetric quivers)
- 'edge_diff'– edges added/deleted (works with skew-symmetric quivers)
- 'green_vertices'– list of green vertices (works with principals)
- 'green_vertices_diff'– green vertices added/removed (works with principals)
- 'red_vertices'– list of red vertices (works with principals)
- 'red_vertices_diff'– red vertices added/removed (works with principals)
- 'urban_renewals'– list of urban renewal vertices
- 'urban_renewals_diff'– urban renewal vertices added/removed
- 'sources'– list of source vertices
- 'sources_diff'– source vertices added/removed
- 'sinks'– list of sink vertices
- 'sinks_diff'– sink vertices added/removed
- 'denominators'– list of all denominators of the cluster variables
 - OUTPUT: - Outputs a dictionary indexed by the vertex numbers. Each vertex will itself also be a dictionary with each desired option included as a key in the dictionary. As an example you would get something similar to: - {0: {'edges': 1}, 1: {'edges': 2}}. This represents that if you were to do a mutation at the current seed then mutating at vertex 0 would result in a quiver with 1 edge and mutating at vertex 0 would result in a quiver with 2 edges.- EXAMPLES: - sage: B = [[0, 4, 0, -1],[-4,0, 3, 0],[0, -3, 0, 1],[1, 0, -1, 0]] sage: S = ClusterSeed(matrix(B)); S.mutate([2,3,1,2,1,3,0,2]) sage: S.mutation_analysis() {0: {'d_matrix': [ 0 0 1 0] [ 0 -1 0 0] [ 0 0 0 -1] [-1 0 0 0], 'denominators': [1, 1, x0, 1], 'edge_diff': 6, 'edges': 13, 'green_vertices': [0, 1, 3], 'green_vertices_diff': {'added': [0], 'removed': []}, 'red_vertices': [2], 'red_vertices_diff': {'added': [], 'removed': [0]}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [2]}, 'sources': [], 'sources_diff': {'added': [], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 1: {'d_matrix': [ 1 4 1 0] [ 0 1 0 0] [ 0 0 0 -1] [ 1 4 0 0], 'denominators': [x0*x3, x0^4*x1*x3^4, x0, 1], 'edge_diff': 2, 'edges': 9, 'green_vertices': [0, 3], 'green_vertices_diff': {'added': [0], 'removed': [1]}, 'red_vertices': [1, 2], 'red_vertices_diff': {'added': [1], 'removed': [0]}, 'sinks': [2], 'sinks_diff': {'added': [], 'removed': []}, 'sources': [], 'sources_diff': {'added': [], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 2: {'d_matrix': [ 1 0 0 0] [ 0 -1 0 0] [ 0 0 0 -1] [ 1 0 1 0], 'denominators': [x0*x3, 1, x3, 1], 'edge_diff': 0, 'edges': 7, 'green_vertices': [1, 2, 3], 'green_vertices_diff': {'added': [2], 'removed': []}, 'red_vertices': [0], 'red_vertices_diff': {'added': [], 'removed': [2]}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [2]}, 'sources': [2], 'sources_diff': {'added': [2], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 3: {'d_matrix': [ 1 0 1 1] [ 0 -1 0 0] [ 0 0 0 1] [ 1 0 0 1], 'denominators': [x0*x3, 1, x0, x0*x2*x3], 'edge_diff': -1, 'edges': 6, 'green_vertices': [1], 'green_vertices_diff': {'added': [], 'removed': [3]}, 'red_vertices': [0, 2, 3], 'red_vertices_diff': {'added': [3], 'removed': []}, 'sinks': [2], 'sinks_diff': {'added': [], 'removed': []}, 'sources': [1], 'sources_diff': {'added': [1], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}} sage: S = ClusterSeed(['A',3]).principal_extension() sage: S.mutation_analysis() {0: {'d_matrix': [ 1 0 0] [ 0 -1 0] [ 0 0 -1], 'denominators': [x0, 1, 1], 'green_vertices': [1, 2], 'green_vertices_diff': {'added': [], 'removed': [0]}, 'red_vertices': [0], 'red_vertices_diff': {'added': [0], 'removed': []}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [1]}, 'sources': [4, 5], 'sources_diff': {'added': [], 'removed': [3]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 1: {'d_matrix': [-1 0 0] [ 0 1 0] [ 0 0 -1], 'denominators': [1, x1, 1], 'green_vertices': [0, 2], 'green_vertices_diff': {'added': [], 'removed': [1]}, 'red_vertices': [1], 'red_vertices_diff': {'added': [1], 'removed': []}, 'sinks': [0, 2, 4], 'sinks_diff': {'added': [0, 2, 4], 'removed': [1]}, 'sources': [1, 3, 5], 'sources_diff': {'added': [1], 'removed': [4]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 2: {'d_matrix': [-1 0 0] [ 0 -1 0] [ 0 0 1], 'denominators': [1, 1, x2], 'green_vertices': [0, 1], 'green_vertices_diff': {'added': [], 'removed': [2]}, 'red_vertices': [2], 'red_vertices_diff': {'added': [2], 'removed': []}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [1]}, 'sources': [3, 4], 'sources_diff': {'added': [], 'removed': [5]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}} - >>> from sage.all import * >>> B = [[Integer(0), Integer(4), Integer(0), -Integer(1)],[-Integer(4),Integer(0), Integer(3), Integer(0)],[Integer(0), -Integer(3), Integer(0), Integer(1)],[Integer(1), Integer(0), -Integer(1), Integer(0)]] >>> S = ClusterSeed(matrix(B)); S.mutate([Integer(2),Integer(3),Integer(1),Integer(2),Integer(1),Integer(3),Integer(0),Integer(2)]) >>> S.mutation_analysis() {0: {'d_matrix': [ 0 0 1 0] [ 0 -1 0 0] [ 0 0 0 -1] [-1 0 0 0], 'denominators': [1, 1, x0, 1], 'edge_diff': 6, 'edges': 13, 'green_vertices': [0, 1, 3], 'green_vertices_diff': {'added': [0], 'removed': []}, 'red_vertices': [2], 'red_vertices_diff': {'added': [], 'removed': [0]}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [2]}, 'sources': [], 'sources_diff': {'added': [], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 1: {'d_matrix': [ 1 4 1 0] [ 0 1 0 0] [ 0 0 0 -1] [ 1 4 0 0], 'denominators': [x0*x3, x0^4*x1*x3^4, x0, 1], 'edge_diff': 2, 'edges': 9, 'green_vertices': [0, 3], 'green_vertices_diff': {'added': [0], 'removed': [1]}, 'red_vertices': [1, 2], 'red_vertices_diff': {'added': [1], 'removed': [0]}, 'sinks': [2], 'sinks_diff': {'added': [], 'removed': []}, 'sources': [], 'sources_diff': {'added': [], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 2: {'d_matrix': [ 1 0 0 0] [ 0 -1 0 0] [ 0 0 0 -1] [ 1 0 1 0], 'denominators': [x0*x3, 1, x3, 1], 'edge_diff': 0, 'edges': 7, 'green_vertices': [1, 2, 3], 'green_vertices_diff': {'added': [2], 'removed': []}, 'red_vertices': [0], 'red_vertices_diff': {'added': [], 'removed': [2]}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [2]}, 'sources': [2], 'sources_diff': {'added': [2], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 3: {'d_matrix': [ 1 0 1 1] [ 0 -1 0 0] [ 0 0 0 1] [ 1 0 0 1], 'denominators': [x0*x3, 1, x0, x0*x2*x3], 'edge_diff': -1, 'edges': 6, 'green_vertices': [1], 'green_vertices_diff': {'added': [], 'removed': [3]}, 'red_vertices': [0, 2, 3], 'red_vertices_diff': {'added': [3], 'removed': []}, 'sinks': [2], 'sinks_diff': {'added': [], 'removed': []}, 'sources': [1], 'sources_diff': {'added': [1], 'removed': []}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}} >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.mutation_analysis() {0: {'d_matrix': [ 1 0 0] [ 0 -1 0] [ 0 0 -1], 'denominators': [x0, 1, 1], 'green_vertices': [1, 2], 'green_vertices_diff': {'added': [], 'removed': [0]}, 'red_vertices': [0], 'red_vertices_diff': {'added': [0], 'removed': []}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [1]}, 'sources': [4, 5], 'sources_diff': {'added': [], 'removed': [3]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 1: {'d_matrix': [-1 0 0] [ 0 1 0] [ 0 0 -1], 'denominators': [1, x1, 1], 'green_vertices': [0, 2], 'green_vertices_diff': {'added': [], 'removed': [1]}, 'red_vertices': [1], 'red_vertices_diff': {'added': [1], 'removed': []}, 'sinks': [0, 2, 4], 'sinks_diff': {'added': [0, 2, 4], 'removed': [1]}, 'sources': [1, 3, 5], 'sources_diff': {'added': [1], 'removed': [4]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}, 2: {'d_matrix': [-1 0 0] [ 0 -1 0] [ 0 0 1], 'denominators': [1, 1, x2], 'green_vertices': [0, 1], 'green_vertices_diff': {'added': [], 'removed': [2]}, 'red_vertices': [2], 'red_vertices_diff': {'added': [2], 'removed': []}, 'sinks': [], 'sinks_diff': {'added': [], 'removed': [1]}, 'sources': [3, 4], 'sources_diff': {'added': [], 'removed': [5]}, 'urban_renewals': [], 'urban_renewals_diff': {'added': [], 'removed': []}}} 
 - mutation_class(depth=+Infinity, show_depth=False, return_paths=False, up_to_equivalence=True, only_sink_source=False)[source]¶
- Return the mutation class of - selfwith respect to certain constraints.- Note - Vertex labels are not tracked in this method. - See also - INPUT: - depth– (default:- infinity) integer, only seeds with distance at most depth from- selfare returned
- show_depth– boolean (default:- False); if- True, the actual depth of the mutation is shown
- return_paths– boolean (default:- False); if- True, a shortest path of mutation sequences from self to the given quiver is returned as well
- up_to_equivalence– boolean (default:- True); if- True, only seeds up to equivalence are considered
- sink_source– boolean (default:- False); if- True, only mutations at sinks and sources are applied
 - EXAMPLES: - for examples see - mutation_class_iter()
 
 - mutation_class_iter(depth=+Infinity, show_depth=False, return_paths=False, up_to_equivalence=True, only_sink_source=False)[source]¶
- Return an iterator for the mutation class of - selfwith respect to certain constraints.- INPUT: - depth– (default: infinity) integer or infinity, only seeds with distance at most- depthfrom- selfare returned
- show_depth– boolean (default:- False); if- True, the current depth of the mutation is shown while computing
- return_paths– boolean (default:- False); if- True, a shortest path of mutations from- selfto the given quiver is returned as well
- up_to_equivalence– boolean (default:- True); if- True, only one seed up to simultaneous permutation of rows and columns of the exchange matrix is recorded
- sink_source– boolean (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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> it = S.mutation_class_iter() >>> 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] - >>> from sage.all import * >>> it = S.mutation_class_iter(depth=Integer(1)) >>> 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 - >>> from sage.all import * >>> it = S.mutation_class_iter(show_depth=True) >>> 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: mutation_class = list(it) sage: len(mutation_class) 14 sage: mutation_class[0] (A seed for a cluster algebra of rank 3 of type ['A', 3], []) - >>> from sage.all import * >>> it = S.mutation_class_iter(return_paths=True) >>> mutation_class = list(it) >>> len(mutation_class) 14 >>> mutation_class[Integer(0)] (A seed for a cluster algebra of rank 3 of type ['A', 3], []) - 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: mutation_class = list(it) sage: len(mutation_class) 10 sage: mutation_class[0] (A seed for a cluster algebra of rank 2 of type ['A', 2], []) - >>> from sage.all import * >>> it = S.mutation_class_iter(up_to_equivalence=False) >>> len([T for T in it]) 84 >>> it = ClusterSeed(['A',Integer(2)]).mutation_class_iter(return_paths=True, ... up_to_equivalence=False) >>> mutation_class = list(it) >>> len(mutation_class) 10 >>> mutation_class[Integer(0)] (A seed for a cluster algebra of rank 2 of type ['A', 2], []) - Check that Issue #14638 is fixed: - sage: S = ClusterSeed(['E',6]) sage: MC = S.mutation_class(depth=7); len(MC) # long time 534 - >>> from sage.all import * >>> S = ClusterSeed(['E',Integer(6)]) >>> MC = S.mutation_class(depth=Integer(7)); len(MC) # long time 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]) - >>> from sage.all import * >>> S = ClusterSeed(['A',[Integer(1),Integer(1)],Integer(1)]) >>> it = S.mutation_class_iter() >>> next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] >>> next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] >>> next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] >>> next(it) A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1] >>> it = S.mutation_class_iter(depth=Integer(3), return_paths=True) >>> 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')[source]¶
- Return the seeds obtained by mutating - selfat all vertices in- sequence.- INPUT: - sequence– an iterable of vertices of self
- show_sequence– boolean (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- sequenceof 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(2)]) >>> for T in S.mutation_sequence([Integer(0),Integer(1),Integer(0)]): ... print(T.b_matrix()) [ 0 -1] [ 1 0] [ 0 1] [-1 0] [ 0 -1] [ 1 0] >>> S = ClusterSeed(['A',Integer(2)]) >>> S.mutation_sequence([Integer(0),Integer(1),Integer(0),Integer(1)], return_output='var') [(x1 + 1)/x0, (x0 + x1 + 1)/(x0*x1), (x0 + 1)/x1, x0] 
 - mutation_type()[source]¶
- Return 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 happened 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] sage: S = ClusterSeed(DiGraph([['a','b'],['c','b'],['c','d'],['e','d']]), ....: frozen=['c']) sage: S.mutation_type() [ ['A', 2], ['A', 2] ] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(5)]) >>> S._mutation_type = S._quiver._mutation_type = None >>> S.mutation_type() ['A', 5] >>> S = ClusterSeed([(Integer(0),Integer(1)),(Integer(1),Integer(2)),(Integer(2),Integer(3)),(Integer(3),Integer(4))]) >>> S.mutation_type() ['A', 5] >>> S = ClusterSeed(DiGraph([['a','b'],['c','b'],['c','d'],['e','d']]), ... frozen=['c']) >>> S.mutation_type() [ ['A', 2], ['A', 2] ] 
- 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]] - >>> from sage.all import * >>> S = ClusterSeed(['E',Integer(8),[Integer(1),Integer(1)]]); S A seed for a cluster algebra of rank 10 of type ['E', 8, [1, 1]] >>> S._mutation_type = S._quiver._mutation_type = None; S A seed for a cluster algebra of rank 10 >>> 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] - >>> from sage.all import * >>> S = ClusterSeed(['D',Integer(4),Integer(1)]) >>> S._mutation_type = S._quiver._mutation_type = None >>> 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] - >>> from sage.all import * >>> S = ClusterSeed(['X',Integer(6)]) >>> S._mutation_type = S._quiver._mutation_type = None >>> 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' - >>> from sage.all import * >>> S = ClusterSeed(['GR',[Integer(4),Integer(9)]]) >>> S._mutation_type = S._quiver._mutation_type = None >>> S.mutation_type() 'undetermined infinite mutation type' 
 
 - mutations()[source]¶
- Return the list of mutations - selfhas undergone if they are being tracked.- EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: S.mutations() [] sage: S.mutate([0,1,0,2]) sage: S.mutations() [0, 1, 0, 2] sage: S.track_mutations(False) sage: S.mutations() Traceback (most recent call last): ... ValueError: Not recording mutation sequence. Need to track mutations. - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.mutations() [] >>> S.mutate([Integer(0),Integer(1),Integer(0),Integer(2)]) >>> S.mutations() [0, 1, 0, 2] >>> S.track_mutations(False) >>> S.mutations() Traceback (most recent call last): ... ValueError: Not recording mutation sequence. Need to track mutations. 
 - n()[source]¶
- Return the number of exchangeable variables of - self.- EXAMPLES: - sage: S = ClusterSeed(['A', 3]) sage: S.n() 3 - >>> from sage.all import * >>> S = ClusterSeed(['A', Integer(3)]) >>> S.n() 3 
 - oriented_exchange_graph()[source]¶
- Return the oriented exchange graph of - selfas a directed graph.- The seed must be a cluster seed for a cluster algebra of finite type with principal coefficients (the corresponding quiver must have mutable vertices \(0,1,...,n-1\)). - EXAMPLES: - sage: S = ClusterSeed(['A', 2]).principal_extension() sage: G = S.oriented_exchange_graph(); G Digraph on 5 vertices sage: G.out_degree_sequence() [2, 1, 1, 1, 0] sage: S = ClusterSeed(['B', 2]).principal_extension() sage: G = S.oriented_exchange_graph(); G Digraph on 6 vertices sage: G.out_degree_sequence() [2, 1, 1, 1, 1, 0] - >>> from sage.all import * >>> S = ClusterSeed(['A', Integer(2)]).principal_extension() >>> G = S.oriented_exchange_graph(); G Digraph on 5 vertices >>> G.out_degree_sequence() [2, 1, 1, 1, 0] >>> S = ClusterSeed(['B', Integer(2)]).principal_extension() >>> G = S.oriented_exchange_graph(); G Digraph on 6 vertices >>> G.out_degree_sequence() [2, 1, 1, 1, 1, 0] 
 - plot(circular=False, mark=None, save_pos=False, force_c=False, with_greens=False, add_labels=False)[source]¶
- Return the plot of the quiver of - self.- INPUT: - circular– boolean (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– boolean (default:- False); if- True, the positions of the vertices are saved
- force_c– boolean (default:- False); if- True, will show the frozen vertices even if they were never initialized
- with_greens– boolean (default:- False); if- True, will display the green vertices in green
- add_labels– boolean (default:- False); if- True, will use the initial variables as labels
 - EXAMPLES: - sage: S = ClusterSeed(['A',5]) sage: S.plot() # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives sage: S.plot(circular=True) # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives sage: S.plot(circular=True, mark=1) # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(5)]) >>> S.plot() # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives >>> S.plot(circular=True) # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives >>> S.plot(circular=True, mark=Integer(1)) # needs sage.plot sage.symbolic Graphics object consisting of 15 graphics primitives 
 - principal_extension()[source]¶
- Return the principal extension of - self, yielding a \(2n \times n\) matrix.- Raises an error if the input seed has a non-square exchange matrix. 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: S = ClusterSeed(['A', 4], user_labels=['a', 'b', 'c', 'd']) sage: T = S.principal_extension() sage: T.cluster() [a, b, c, d] sage: T.coefficients() [y0, y1, y2, y3] sage: S2 = ClusterSeed(['A', 4], user_labels={0:'a', 1:'b', 2:'c', 3:'d'}) sage: S2 == S True sage: T2 = S2.principal_extension() sage: T2 == T True - >>> from sage.all import * >>> S = ClusterSeed([[Integer(0),Integer(1)],[Integer(1),Integer(2)],[Integer(2),Integer(3)],[Integer(2),Integer(4)]]); S A seed for a cluster algebra of rank 5 >>> T = S.principal_extension(); T A seed for a cluster algebra of rank 5 with principal coefficients >>> 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] >>> S = ClusterSeed(['A', Integer(4)], user_labels=['a', 'b', 'c', 'd']) >>> T = S.principal_extension() >>> T.cluster() [a, b, c, d] >>> T.coefficients() [y0, y1, y2, y3] >>> S2 = ClusterSeed(['A', Integer(4)], user_labels={Integer(0):'a', Integer(1):'b', Integer(2):'c', Integer(3):'d'}) >>> S2 == S True >>> T2 = S2.principal_extension() >>> T2 == T True 
 - quiver()[source]¶
- Return the quiver associated to - self.- EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: S.quiver() Quiver on 3 vertices of type ['A', 3] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.quiver() Quiver on 3 vertices of type ['A', 3] 
 - red_vertices()[source]¶
- Return the list of red vertices of - self.- A vertex is defined to be red if its c-vector has all nonnegative entries. More information on red vertices can be found at [BDP2013]. - OUTPUT: the red vertices as a list of integers - EXAMPLES: - sage: ClusterSeed(['A',3]).principal_extension().red_vertices() [] sage: ClusterSeed(['A',[3,3],1]).principal_extension().red_vertices() [] sage: Q = ClusterSeed(['A',[3,3],1]).principal_extension() sage: Q.mutate(1) sage: Q.red_vertices() [1] - >>> from sage.all import * >>> ClusterSeed(['A',Integer(3)]).principal_extension().red_vertices() [] >>> ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension().red_vertices() [] >>> Q = ClusterSeed(['A',[Integer(3),Integer(3)],Integer(1)]).principal_extension() >>> Q.mutate(Integer(1)) >>> Q.red_vertices() [1] 
 - reorient(data)[source]¶
- Reorients - selfwith 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 \((i,j)\) for which neither \((i,j)\) nor \((j,i)\) is an edge of - self.
 - INPUT: - data– an iterator defining a total order on- self.vertices(), or an iterator of ordered pairs in- selfdefining 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',[Integer(2),Integer(3)],Integer(1)]) >>> S.mutation_type() ['A', [2, 3], 1] >>> S.reorient([(Integer(0),Integer(1)),(Integer(2),Integer(3))]) >>> S.mutation_type() ['D', 5] >>> S.reorient([(Integer(1),Integer(0)),(Integer(2),Integer(3))]) >>> S.mutation_type() ['A', [1, 4], 1] >>> S.reorient([Integer(0),Integer(1),Integer(2),Integer(3),Integer(4)]) >>> S.mutation_type() ['A', [1, 4], 1] 
 - reset_cluster()[source]¶
- Reset the cluster of - selfto 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] sage: S = ClusterSeed(['B',3], user_labels=[[1,2],[2,3],[3,4]], ....: user_labels_prefix='p') sage: S.mutate([0,1]) sage: S.cluster() [(p_2_3 + 1)/p_1_2, (p_1_2*p_3_4^2 + p_2_3 + 1)/(p_1_2*p_2_3), p_3_4] sage: S.reset_cluster() sage: S.cluster() [p_1_2, p_2_3, p_3_4] sage: S.g_matrix() [1 0 0] [0 1 0] [0 0 1] sage: S.f_polynomials() [1, 1, 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> S.mutate([Integer(1),Integer(2),Integer(1)]) >>> S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] >>> S.reset_cluster() >>> S.cluster() [x0, x1, x2] >>> T = S.principal_extension() >>> T.cluster() [x0, x1, x2] >>> T.mutate([Integer(1),Integer(2),Integer(1)]) >>> T.cluster() [x0, (x1*y2 + x0)/x2, (x1*y1*y2 + x0*y1 + x2)/(x1*x2)] >>> T.reset_cluster() >>> T.cluster() [x0, x1, x2] >>> S = ClusterSeed(['B',Integer(3)], user_labels=[[Integer(1),Integer(2)],[Integer(2),Integer(3)],[Integer(3),Integer(4)]], ... user_labels_prefix='p') >>> S.mutate([Integer(0),Integer(1)]) >>> S.cluster() [(p_2_3 + 1)/p_1_2, (p_1_2*p_3_4^2 + p_2_3 + 1)/(p_1_2*p_2_3), p_3_4] >>> S.reset_cluster() >>> S.cluster() [p_1_2, p_2_3, p_3_4] >>> S.g_matrix() [1 0 0] [0 1 0] [0 0 1] >>> S.f_polynomials() [1, 1, 1] 
 - reset_coefficients()[source]¶
- Reset the coefficients of - selfto the frozen variables but keep the current cluster.- This raises an error if the number of frozen variables is different from the number of exchangeable variables. - Warning - This command to be phased out since - use_c_vectors()does this more effectively.- 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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]).principal_extension() >>> S.b_matrix() [ 0 1 0] [-1 0 -1] [ 0 1 0] [ 1 0 0] [ 0 1 0] [ 0 0 1] >>> S.mutate([Integer(1),Integer(2),Integer(1)]) >>> S.b_matrix() [ 0 1 -1] [-1 0 1] [ 1 -1 0] [ 1 0 0] [ 0 1 -1] [ 0 0 -1] >>> S.reset_coefficients() >>> 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)[source]¶
- Save the plot of the underlying digraph of the quiver of - self.- INPUT: - filename– the filename the image is saved to
- circular– boolean (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– boolean (default:- False); if- True, the positions of the vertices are saved
 - EXAMPLES: - sage: S = ClusterSeed(['F',4,[1,2]]) sage: import tempfile sage: with tempfile.NamedTemporaryFile(suffix='.png') as f: # needs sage.plot sage.symbolic ....: S.save_image(f.name) - >>> from sage.all import * >>> S = ClusterSeed(['F',Integer(4),[Integer(1),Integer(2)]]) >>> import tempfile >>> with tempfile.NamedTemporaryFile(suffix='.png') as f: # needs sage.plot sage.symbolic ... S.save_image(f.name) 
 - set_c_matrix(data)[source]¶
- Will force set the c-matrix according to a matrix, a quiver, or a seed. - INPUT: - data– the matrix to set the c-matrix to; also allowed to be a quiver or cluster seed, in which case the b-matrix is used
 - EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: X = matrix([[0,0,1],[0,1,0],[1,0,0]]) sage: S.set_c_matrix(X) sage: S.c_matrix() [0 0 1] [0 1 0] [1 0 0] sage: Y = matrix([[-1,0,1],[0,1,0],[1,0,0]]) sage: S.set_c_matrix(Y) C matrix does not look to be valid - there exists a column containing positive and negative entries. Continuing... sage: Z = matrix([[1,0,1],[0,1,0],[2,0,2]]) sage: S.set_c_matrix(Z) C matrix does not look to be valid - not a linearly independent set. Continuing... - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> X = matrix([[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)]]) >>> S.set_c_matrix(X) >>> S.c_matrix() [0 0 1] [0 1 0] [1 0 0] >>> Y = matrix([[-Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)]]) >>> S.set_c_matrix(Y) C matrix does not look to be valid - there exists a column containing positive and negative entries. Continuing... >>> Z = matrix([[Integer(1),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(2),Integer(0),Integer(2)]]) >>> S.set_c_matrix(Z) C matrix does not look to be valid - not a linearly independent set. Continuing... 
 - set_cluster(cluster, force=False)[source]¶
- Set the cluster for - selfto- cluster.- Warning - Initialization may lead to inconsistent data. - INPUT: - cluster– an iterable defining a cluster for- self
 - 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: cluster2 = S.cluster() sage: S.set_cluster(cluster) Warning: using set_cluster at this point could lead to inconsistent seed data. sage: S.set_cluster(cluster, force=True) sage: S.cluster() [x0, x1, x2] sage: S.set_cluster(cluster2, force=True) sage: S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] sage: S = ClusterSeed(['A',3]); S.use_fpolys(False) sage: S.set_cluster([1,1,1]) Warning: clusters not being tracked so this command is ignored. - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> cluster = S.cluster() >>> S.mutate([Integer(1),Integer(2),Integer(1)]) >>> S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] >>> cluster2 = S.cluster() >>> S.set_cluster(cluster) Warning: using set_cluster at this point could lead to inconsistent seed data. >>> S.set_cluster(cluster, force=True) >>> S.cluster() [x0, x1, x2] >>> S.set_cluster(cluster2, force=True) >>> S.cluster() [x0, (x1 + 1)/x2, (x0*x2 + x1 + 1)/(x1*x2)] >>> S = ClusterSeed(['A',Integer(3)]); S.use_fpolys(False) >>> S.set_cluster([Integer(1),Integer(1),Integer(1)]) Warning: clusters not being tracked so this command is ignored. 
 - show(fig_size=1, circular=False, mark=None, save_pos=False, force_c=False, with_greens=False, add_labels=False)[source]¶
- Shows the plot of the quiver of - self.- INPUT: - fig_size– (default: 1) factor by which the size of the plot is multiplied
- circular– boolean (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– boolean (default:- False); if- True, the positions of the vertices are saved
- force_c– boolean (default:- False); if- True, will show the frozen vertices even if they were never initialized
- with_greens– boolean (default:- False); if- True, will display the green vertices in green
- add_labels– boolean (default:- False); if- True, will use the initial variables as labels
 
 - smallest_c_vector()[source]¶
- Return the vertex with the smallest c-vector. - OUTPUT: integer - EXAMPLES: - sage: B = matrix([[0,2], [-2,0]]) sage: C = ClusterSeed(B).principal_extension() sage: C.mutate(0) sage: C.smallest_c_vector() 0 - >>> from sage.all import * >>> B = matrix([[Integer(0),Integer(2)], [-Integer(2),Integer(0)]]) >>> C = ClusterSeed(B).principal_extension() >>> C.mutate(Integer(0)) >>> C.smallest_c_vector() 0 
 - track_mutations(use=True)[source]¶
- Begin tracking the mutation path. - Warning - May initialize all other data to ensure that all c-, d-, and g-vectors agree on the start of mutations. - INPUT: - use– boolean (default:- True); if- True, will begin filling the mutation path
 - EXAMPLES: - sage: S = ClusterSeed(['A',4]); S.track_mutations(False) sage: S.mutate(0) sage: S.mutations() Traceback (most recent call last): ... ValueError: Not recording mutation sequence. Need to track mutations. sage: S.track_mutations(True) sage: S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: S.mutate([0,1]) sage: S.mutations() [0, 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]); S.track_mutations(False) >>> S.mutate(Integer(0)) >>> S.mutations() Traceback (most recent call last): ... ValueError: Not recording mutation sequence. Need to track mutations. >>> S.track_mutations(True) >>> S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] >>> S.mutate([Integer(0),Integer(1)]) >>> S.mutations() [0, 1] 
 - universal_extension()[source]¶
- Return the universal extension of - self.- This is the initial seed of the associated cluster algebra with universal coefficients, as defined in section 12 of [FZ2007]. - This method works only if - selfis 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] sage: S = ClusterSeed(['A', 5], user_labels=[-2, -1, 0, 1 ,2]) sage: U = S.universal_extension() sage: U.b_matrix() == ClusterSeed(['A', 5]).universal_extension().b_matrix() True - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(2)]) >>> T = S.universal_extension() >>> T.b_matrix() [ 0 1] [-1 0] [-1 0] [ 1 0] [ 1 -1] [ 0 1] [ 0 -1] >>> S = ClusterSeed(['A',Integer(3)]) >>> T = S.universal_extension() >>> 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] >>> S = ClusterSeed(['B',Integer(2)]) >>> T = S.universal_extension() >>> T.b_matrix() [ 0 1] [-2 0] [-1 0] [ 1 0] [ 1 -1] [ 2 -1] [ 0 1] [ 0 -1] >>> S = ClusterSeed(['A', Integer(5)], user_labels=[-Integer(2), -Integer(1), Integer(0), Integer(1) ,Integer(2)]) >>> U = S.universal_extension() >>> U.b_matrix() == ClusterSeed(['A', Integer(5)]).universal_extension().b_matrix() True 
 - urban_renewals(return_first=False)[source]¶
- Return the list of the urban renewal vertices of - self.- An urban renewal vertex is one in which there are two arrows pointing toward the vertex and two arrows pointing away. - INPUT: - return_first– boolean (default:- False); if- True, will return the first urban renewal
 - OUTPUT: - A list of vertices (as integers) - EXAMPLES: - sage: G = ClusterSeed(['GR',[4,9]]); G.urban_renewals() [5, 6] - >>> from sage.all import * >>> G = ClusterSeed(['GR',[Integer(4),Integer(9)]]); G.urban_renewals() [5, 6] 
 - use_c_vectors(use=True, bot_is_c=False, force=False)[source]¶
- Reconstruct \(c\)-vectors from other data or initialize if no usable data exists. - Warning: Initialization may lead to inconsistent data. - INPUT: - use– boolean (default:- True); if- True, will use \(c\)-vectors
- bot_is_c– boolean (default:- False); if- Trueand- ClusterSeed- selfhas- self._m == self._n, then will assume bottom half of the extended exchange matrix is the c-matrix. If- True, lets the- ClusterSeedknow c-vectors can be calculated.
 - EXAMPLES: - sage: S = ClusterSeed(['A',4]) sage: S.use_c_vectors(False); S.use_g_vectors(False) sage: S.use_fpolys(False); S.track_mutations(False) sage: S.use_c_vectors(True) Warning: Initializing c-vectors at this point could lead to inconsistent seed data. sage: S.use_c_vectors(True, force=True) sage: S.c_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: S = ClusterSeed(['A',4]) sage: S.use_c_vectors(False); S.use_g_vectors(False) sage: S.use_fpolys(False); S.track_mutations(False) sage: S.mutate(1) sage: S.use_c_vectors(True, force=True) sage: S.c_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_c_vectors(False); S.use_g_vectors(False) >>> S.use_fpolys(False); S.track_mutations(False) >>> S.use_c_vectors(True) Warning: Initializing c-vectors at this point could lead to inconsistent seed data. >>> S.use_c_vectors(True, force=True) >>> S.c_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_c_vectors(False); S.use_g_vectors(False) >>> S.use_fpolys(False); S.track_mutations(False) >>> S.mutate(Integer(1)) >>> S.use_c_vectors(True, force=True) >>> S.c_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] 
 - use_d_vectors(use=True, force=False)[source]¶
- Reconstruct \(d\)-vectors from other data or initialize if no usable data exists. - Warning - Initialization may lead to inconsistent data. - INPUT: - use– boolean (default:- True); if- True, will use \(d\)-vectors
 - EXAMPLES: - sage: S = ClusterSeed(['A',4]) sage: S.use_d_vectors(True) sage: S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S = ClusterSeed(['A',4]); S.use_d_vectors(False) sage: S.track_mutations(False); S.mutate(1); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S.use_fpolys(False) sage: S.d_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate d-vectors. Need to use d vectors. sage: S = ClusterSeed(['A',4]); S.use_d_vectors(False) sage: S.track_mutations(False); S.mutate(1); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S.use_fpolys(False) sage: S.use_d_vectors(True) Warning: Initializing d-vectors at this point could lead to inconsistent seed data. sage: S.use_d_vectors(True, force=True) sage: S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S = ClusterSeed(['A',4]); S.mutate(1); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] sage: S = ClusterSeed(['A',4]) sage: S.use_d_vectors(True); S.mutate(1); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_d_vectors(True) >>> S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S = ClusterSeed(['A',Integer(4)]); S.use_d_vectors(False) >>> S.track_mutations(False); S.mutate(Integer(1)); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S.use_fpolys(False) >>> S.d_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate d-vectors. Need to use d vectors. >>> S = ClusterSeed(['A',Integer(4)]); S.use_d_vectors(False) >>> S.track_mutations(False); S.mutate(Integer(1)); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S.use_fpolys(False) >>> S.use_d_vectors(True) Warning: Initializing d-vectors at this point could lead to inconsistent seed data. >>> S.use_d_vectors(True, force=True) >>> S.d_matrix() [-1 0 0 0] [ 0 -1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S = ClusterSeed(['A',Integer(4)]); S.mutate(Integer(1)); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_d_vectors(True); S.mutate(Integer(1)); S.d_matrix() [-1 0 0 0] [ 0 1 0 0] [ 0 0 -1 0] [ 0 0 0 -1] 
 - use_fpolys(use=True, user_labels=None, user_labels_prefix=None)[source]¶
- Use \(F\)-polynomials in our Cluster Seed. - Note: This will automatically try to recompute the cluster variables if possible - INPUT: - use– boolean (default:- True); if- True, will use \(F\)-polynomials
- user_labels– (default:- None) if set, will overwrite the default cluster variable- labels
- user_labels_prefix– (default:- None) if set, will overwrite the default
 - EXAMPLES: - sage: S = ClusterSeed(['A',4]); S.use_fpolys(False); S._cluster sage: S.use_fpolys(True) sage: S.cluster() [x0, x1, x2, x3] sage: S = ClusterSeed(['A',4]); S.use_fpolys(False); S.track_mutations(False) sage: S.mutate(1) sage: S.use_fpolys(True) Traceback (most recent call last): ... ValueError: F-polynomials and Cluster Variables cannot be reconstructed from given data. sage: S.cluster() Traceback (most recent call last): ... ValueError: Clusters not being tracked - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]); S.use_fpolys(False); S._cluster >>> S.use_fpolys(True) >>> S.cluster() [x0, x1, x2, x3] >>> S = ClusterSeed(['A',Integer(4)]); S.use_fpolys(False); S.track_mutations(False) >>> S.mutate(Integer(1)) >>> S.use_fpolys(True) Traceback (most recent call last): ... ValueError: F-polynomials and Cluster Variables cannot be reconstructed from given data. >>> S.cluster() Traceback (most recent call last): ... ValueError: Clusters not being tracked 
 - use_g_vectors(use=True, force=False)[source]¶
- Reconstruct g-vectors from other data or initialize if no usable data exists. - Warning - Initialization may lead to inconsistent data. - INPUT: - use– boolean (default:- True); if- True, will use g-vectors
 - EXAMPLES: - sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False) sage: S.use_g_vectors(True) sage: S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False) sage: S.mutate(1) sage: S.use_g_vectors(True) sage: S.g_matrix() [ 1 0 0 0] [ 0 -1 0 0] [ 0 0 1 0] [ 0 0 0 1] sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False); S.track_mutations(False) sage: S.mutate(1) sage: S.use_c_vectors(False) sage: S.g_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate g-vectors. Need to use g vectors. sage: S = ClusterSeed(['A',4]) sage: S.use_g_vectors(False); S.use_fpolys(False); S.track_mutations(False) sage: S.mutate(1) sage: S.use_c_vectors(False) sage: S.use_g_vectors(True) Warning: Initializing g-vectors at this point could lead to inconsistent seed data. sage: S.use_g_vectors(True, force=True) sage: S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False) >>> S.use_g_vectors(True) >>> S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False) >>> S.mutate(Integer(1)) >>> S.use_g_vectors(True) >>> S.g_matrix() [ 1 0 0 0] [ 0 -1 0 0] [ 0 0 1 0] [ 0 0 0 1] >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False); S.track_mutations(False) >>> S.mutate(Integer(1)) >>> S.use_c_vectors(False) >>> S.g_matrix() Traceback (most recent call last): ... ValueError: Unable to calculate g-vectors. Need to use g vectors. >>> S = ClusterSeed(['A',Integer(4)]) >>> S.use_g_vectors(False); S.use_fpolys(False); S.track_mutations(False) >>> S.mutate(Integer(1)) >>> S.use_c_vectors(False) >>> S.use_g_vectors(True) Warning: Initializing g-vectors at this point could lead to inconsistent seed data. >>> S.use_g_vectors(True, force=True) >>> S.g_matrix() [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] 
 - variable_class(depth=+Infinity, ignore_bipartite_belt=False)[source]¶
- Return all cluster variables in the mutation class of - self.- INPUT: - depth– (default: infinity) integer, only seeds with distance at most- depthfrom- selfare returned
- ignore_bipartite_belt– boolean (default:- False); if- True, the algorithm does not use the bipartite belt
 - EXAMPLES: - for examples see - variable_class_iter()
 
 - variable_class_iter(depth=+Infinity, ignore_bipartite_belt=False)[source]¶
- Return an iterator for all cluster variables in the mutation class of - self.- INPUT: - depth– (default: infinity) integer, only seeds with distance at most- depthfrom- selfare returned
- ignore_bipartite_belt– boolean (default:- False); if- True, the algorithm 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) - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> it = S.variable_class_iter() >>> 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) - >>> from sage.all import * >>> it = S.variable_class_iter(depth=Integer(1)) >>> 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 - >>> from sage.all import * >>> it = S.variable_class_iter(depth=Integer(1), ignore_bipartite_belt=True) >>> for T in it: print(T) x0 x1 x2 (x1 + 1)/x2 (x0*x2 + 1)/x1 (x1 + 1)/x0 >>> S.mutate([Integer(0),Integer(1)]) >>> it2 = S.variable_class_iter(depth=Integer(1)) >>> 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) - >>> from sage.all import * >>> S = ClusterSeed(['A',[Integer(1),Integer(1)],Integer(1)]) >>> it = S.variable_class_iter(depth=Integer(2)) >>> 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)[source]¶
- Return the \(k\) -th initial cluster variable for the associated cluster seed, or the cluster variable of the corresponding vertex in - self.quiver.- 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 sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') sage: S = ClusterSeed(dg, frozen=['c']) sage: S.x(0) a sage: S.x('a') a - >>> from sage.all import * >>> S = ClusterSeed(['A', Integer(3)]) >>> S.mutate([Integer(2), Integer(1)]) >>> S.x(Integer(0)) x0 >>> S.x(Integer(1)) x1 >>> S.x(Integer(2)) x2 >>> dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') >>> S = ClusterSeed(dg, frozen=['c']) >>> S.x(Integer(0)) a >>> S.x('a') a 
 - y(k)[source]¶
- Return the \(k\) -th initial coefficient (frozen variable) for the associated cluster seed, or the cluster variable of the corresponding vertex in - self.quiver.- 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 sage: dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') sage: S = ClusterSeed(dg, frozen=['c']) sage: S.y(0) c sage: S.y('c') c - >>> from sage.all import * >>> S = ClusterSeed(['A', Integer(3)]).principal_extension() >>> S.mutate([Integer(2), Integer(1)]) >>> S.y(Integer(0)) y0 >>> S.y(Integer(1)) y1 >>> S.y(Integer(2)) y2 >>> dg = DiGraph([['a', 'b'], ['b', 'c']], format='list_of_edges') >>> S = ClusterSeed(dg, frozen=['c']) >>> S.y(Integer(0)) c >>> S.y('c') c 
 
- class sage.combinat.cluster_algebra_quiver.cluster_seed.ClusterVariable(parent, numerator, denominator, coerce=True, reduce=True, mutation_type=None, variable_type=None, xdim=0)[source]¶
- Bases: - 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("{} {}".format(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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> for T in S.variable_class_iter(): ... print("{} {}".format(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()[source]¶
- Return the almost positive root associated to - selfif- selfis of finite type.- EXAMPLES: - sage: S = ClusterSeed(['A',3]) sage: for T in S.variable_class_iter(): ....: print("{} {}".format(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] - >>> from sage.all import * >>> S = ClusterSeed(['A',Integer(3)]) >>> for T in S.variable_class_iter(): ... print("{} {}".format(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)[source]¶
- Encode a maximal Dyck path from \((0,0)\) to \((n,m)\) (for \(n \geq m \geq 0\)) 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} - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import PathSubset >>> PathSubset(Integer(4),Integer(0)) {1, 3, 5, 7} >>> PathSubset(Integer(4),Integer(1)) {1, 3, 5, 6, 7} >>> PathSubset(Integer(4),Integer(2)) {1, 2, 3, 5, 6, 7} >>> PathSubset(Integer(4),Integer(3)) {1, 2, 3, 4, 5, 6, 7} >>> PathSubset(Integer(4),Integer(4)) {0, 1, 2, 3, 4, 5, 6, 7} 
- sage.combinat.cluster_algebra_quiver.cluster_seed.SetToPath(T)[source]¶
- Rearrange 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] - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import PathSubset >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import SetToPath >>> SetToPath(PathSubset(Integer(4),Integer(0))) [1, 3, 5, 7] >>> SetToPath(PathSubset(Integer(4),Integer(1))) [1, 3, 5, 7, 6] >>> SetToPath(PathSubset(Integer(4),Integer(2))) [1, 3, 2, 5, 7, 6] >>> SetToPath(PathSubset(Integer(4),Integer(3))) [1, 3, 2, 5, 4, 7, 6] >>> SetToPath(PathSubset(Integer(4),Integer(4))) [1, 0, 3, 2, 5, 4, 7, 6] 
- sage.combinat.cluster_algebra_quiver.cluster_seed.coeff_recurs(p, q, a1, a2, b, c)[source]¶
- 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 - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import coeff_recurs >>> coeff_recurs(Integer(1), Integer(1), Integer(5), Integer(5), Integer(3), Integer(3)) 10 
- sage.combinat.cluster_algebra_quiver.cluster_seed.get_green_vertices(C)[source]¶
- Get the green vertices from a matrix. - Will go through each column and return the ones where no entry is greater than 0. - INPUT: - C– the C-matrix to check
 - EXAMPLES: - sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import get_green_vertices sage: S = ClusterSeed(['A',4]); S.mutate([1,2,3,2,0,1,2,0,3]) sage: get_green_vertices(S.c_matrix()) [0, 3] - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import get_green_vertices >>> S = ClusterSeed(['A',Integer(4)]); S.mutate([Integer(1),Integer(2),Integer(3),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0),Integer(3)]) >>> get_green_vertices(S.c_matrix()) [0, 3] 
- sage.combinat.cluster_algebra_quiver.cluster_seed.get_red_vertices(C)[source]¶
- Get the red vertices from a matrix. - Will go through each column and return the ones where no entry is less than 0. - INPUT: - C– the C-matrix to check
 - EXAMPLES: - sage: from sage.combinat.cluster_algebra_quiver.cluster_seed import get_red_vertices sage: S = ClusterSeed(['A',4]); S.mutate([1,2,3,2,0,1,2,0,3]) sage: get_red_vertices(S.c_matrix()) [1, 2] - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import get_red_vertices >>> S = ClusterSeed(['A',Integer(4)]); S.mutate([Integer(1),Integer(2),Integer(3),Integer(2),Integer(0),Integer(1),Integer(2),Integer(0),Integer(3)]) >>> get_red_vertices(S.c_matrix()) [1, 2] 
- sage.combinat.cluster_algebra_quiver.cluster_seed.is_LeeLiZel_allowable(T, n, m, b, c)[source]¶
- 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 [LLZ2014]. - 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 - >>> from sage.all import * >>> from sage.combinat.cluster_algebra_quiver.cluster_seed import is_LeeLiZel_allowable >>> is_LeeLiZel_allowable({Integer(1),Integer(3),Integer(2),Integer(5),Integer(7),Integer(6)},Integer(4),Integer(2),Integer(6),Integer(6)) False >>> is_LeeLiZel_allowable({Integer(1),Integer(2),Integer(5)},Integer(3),Integer(3),Integer(1),Integer(1)) True