{{{id=101| B = Matrix([[0,1],[-1,0]]); B /// [ 0 1] [-1 0] }}} {{{id=102| ClusterSeed? ///

File: /Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py

Type: <type ‘type’>

Definition: ClusterSeed( [noargspec] )

Docstring:

The cluster seed associated to an exchange matrix.

INPUT:

EXAMPLES:

sage: S = ClusterSeed(['A',5]); S
A seed for a cluster algebra of rank 5 of type ['A', 5]

sage: S = ClusterSeed(['A',[2,5],1]); S
A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1]

sage: T = ClusterSeed( S ); T
A seed for a cluster algebra of rank 7 of type ['A', [2, 5], 1]

sage: T = ClusterSeed( S._M ); T
A seed for a cluster algebra of rank 7

sage: T = ClusterSeed( S.quiver()._digraph ); T
A seed for a cluster algebra of rank 7

sage: T = ClusterSeed( S.quiver()._digraph.edges() ); T
A seed for a cluster algebra of rank 7
}}} {{{id=103| S = ClusterSeed(B); S /// A seed for a cluster algebra of rank 2 }}} {{{id=104| S.mutation_type() /// ['A', 2] }}} {{{id=105| S /// A seed for a cluster algebra of rank 2 of type ['A', 2] }}} {{{id=155| /// }}} {{{id=106| S.is_finite() /// True }}} {{{id=107| S.is_mutation_finite() /// True }}} {{{id=109| S.is_acyclic() /// True }}} {{{id=108| S.is_bipartite() /// True }}} {{{id=112| S.show() /// }}} {{{id=113| S.cluster() /// [x0, x1] }}} {{{id=111| S.mutate(0); S.cluster() /// [(x1 + 1)/x0, x1] }}} {{{id=114| S.mutate(1); S.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1} + 1}{x_{0}}, \frac{x_{0} + x_{1} + 1}{x_{0} x_{1}}\right] }}} {{{id=116| S.mutate([0,1]); S.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{0} + 1}{x_{1}}, x_{0}\right] }}} {{{id=117| S.variable_class() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}, \frac{x_{1} + 1}{x_{0}}, \frac{x_{0} + 1}{x_{1}}, \frac{x_{0} + x_{1} + 1}{x_{0} x_{1}}\right] }}} {{{id=156| S.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1} + 1}{x_{0}}, \frac{x_{0} + x_{1} + 1}{x_{0} x_{1}}\right] }}} {{{id=115| S.reset_cluster() /// }}} {{{id=118| SP = S.principal_extension(); SP /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 2 of type ['A', 2] with 2 frozen variables} }}} {{{id=119| SP.b_matrix(); SP.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}\right) \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}, y_{0}, y_{1}\right] }}} {{{id=120| SP.variable_class() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}, y_{0}, y_{1}, \frac{x_{1} + y_{0}}{x_{0}}, \frac{x_{0} y_{1} + 1}{x_{1}}, \frac{x_{0} y_{0} y_{1} + x_{1} + y_{0}}{x_{0} x_{1}}\right] }}} {{{id=157| S.set_cluster([S.x(0)+S.x(1),S.x(0)^2+1,S.y(0),S.y(1)]) /// Traceback (most recent call last): File "", line 1, in File "_sage_input_28.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("Uy5zZXRfY2x1c3RlcihbUy54KDApK1MueCgxKSxTLngoMCleMisxLFMueSgwKSxTLnkoMSldKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/tmp/tmpFB2KL5/___code___.py", line 3, in exec compile(u'S.set_cluster([S.x(_sage_const_0 )+S.x(_sage_const_1 ),S.x(_sage_const_0 )**_sage_const_2 +_sage_const_1 ,S.y(_sage_const_0 ),S.y(_sage_const_1 )])' + '\n', '', 'single') File "", line 1, in File "/Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py", line 427, in y raise ValueError, "The input is not in an index of a frozen variable." ValueError: The input is not in an index of a frozen variable. }}} {{{id=121| SP.mutation_sequence([0,1,0,1,0],return_output='Mat') /// Traceback (most recent call last): File "", line 1, in File "_sage_input_43.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("U1AubXV0YXRpb25fc2VxdWVuY2UoWzAsMSwwLDEsMF0scmV0dXJuX291dHB1dD0nTWF0Jyk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/tmp/tmpK447pj/___code___.py", line 3, in exec compile(u"SP.mutation_sequence([_sage_const_0 ,_sage_const_1 ,_sage_const_0 ,_sage_const_1 ,_sage_const_0 ],return_output='Mat')" + '\n', '', 'single') File "", line 1, in File "/Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py", line 689, in mutation_sequence raise ValueError, 'The parameter `return_output` can only be `seed`, `matrix`, or `var`.' ValueError: The parameter `return_output` can only be `seed`, `matrix`, or `var`. }}} {{{id=122| SP.mutation_sequence([0,1,0,1,0],return_output='matrix') /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \\ 1 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ -1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \\ 0 & -1 \\ 1 & -1 \end{array}\right), \left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ 0 & -1 \\ -1 & 0 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \\ 0 & 1 \\ -1 & 0 \end{array}\right), \left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array}\right)\right] }}} {{{id=123| Fpolys = SP.variable_class(); Fpolys /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[y_{0} y_{1} + y_{0} + 1, y_{0} + 1, y_{0}, y_{1} + 1, y_{1}, 1, 1\right] }}} {{{id=125| SP.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}, 0, 0\right] }}} {{{id=124| SP.set_cluster([1,1,SP.y(0),SP.y(1)]); SP.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, y_{0}, y_{1}\right] }}} {{{id=127| /// }}} {{{id=1| S2 = ClusterSeed(['A',[1,1],1]); S2 /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 2 of type ['A', [1, 1], 1]} }}} {{{id=110| S2.b_matrix() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right) }}} {{{id=98| S2.show() /// }}} {{{id=3| S2.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}\right] }}} {{{id=4| S2.mutate([0,1]); S2.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1}^{2} + 1}{x_{0}}, \frac{x_{1}^{4} + x_{0}^{2} + 2 x_{1}^{2} + 1}{x_{0}^{2} x_{1}}\right] }}} {{{id=5| S2.mutate([0,1]); S2.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1}^{6} + x_{0}^{4} + 2 x_{0}^{2} x_{1}^{2} + 3 x_{1}^{4} + 2 x_{0}^{2} + 3 x_{1}^{2} + 1}{x_{0}^{3} x_{1}^{2}}, \frac{x_{1}^{8} + x_{0}^{6} + 2 x_{0}^{4} x_{1}^{2} + 3 x_{0}^{2} x_{1}^{4} + 4 x_{1}^{6} + 3 x_{0}^{4} + 6 x_{0}^{2} x_{1}^{2} + 6 x_{1}^{4} + 3 x_{0}^{2} + 4 x_{1}^{2} + 1}{x_{0}^{4} x_{1}^{3}}\right] }}} {{{id=6| S2.mutate([0,1]); S2.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1}^{10} + x_{0}^{8} + 2 x_{0}^{6} x_{1}^{2} + 3 x_{0}^{4} x_{1}^{4} + 4 x_{0}^{2} x_{1}^{6} + 5 x_{1}^{8} + 4 x_{0}^{6} + 9 x_{0}^{4} x_{1}^{2} + 12 x_{0}^{2} x_{1}^{4} + 10 x_{1}^{6} + 6 x_{0}^{4} + 12 x_{0}^{2} x_{1}^{2} + 10 x_{1}^{4} + 4 x_{0}^{2} + 5 x_{1}^{2} + 1}{x_{0}^{5} x_{1}^{4}}, \frac{x_{1}^{12} + x_{0}^{10} + 2 x_{0}^{8} x_{1}^{2} + 3 x_{0}^{6} x_{1}^{4} + 4 x_{0}^{4} x_{1}^{6} + 5 x_{0}^{2} x_{1}^{8} + 6 x_{1}^{10} + 5 x_{0}^{8} + 12 x_{0}^{6} x_{1}^{2} + 18 x_{0}^{4} x_{1}^{4} + 20 x_{0}^{2} x_{1}^{6} + 15 x_{1}^{8} + 10 x_{0}^{6} + 24 x_{0}^{4} x_{1}^{2} + 30 x_{0}^{2} x_{1}^{4} + 20 x_{1}^{6} + 10 x_{0}^{4} + 20 x_{0}^{2} x_{1}^{2} + 15 x_{1}^{4} + 5 x_{0}^{2} + 6 x_{1}^{2} + 1}{x_{0}^{6} x_{1}^{5}}\right] }}} {{{id=131| latex( S2.cluster() ) /// \left[\frac{x_{1}^{10} + x_{0}^{8} + 2 x_{0}^{6} x_{1}^{2} + 3 x_{0}^{4} x_{1}^{4} + 4 x_{0}^{2} x_{1}^{6} + 5 x_{1}^{8} + 4 x_{0}^{6} + 9 x_{0}^{4} x_{1}^{2} + 12 x_{0}^{2} x_{1}^{4} + 10 x_{1}^{6} + 6 x_{0}^{4} + 12 x_{0}^{2} x_{1}^{2} + 10 x_{1}^{4} + 4 x_{0}^{2} + 5 x_{1}^{2} + 1}{x_{0}^{5} x_{1}^{4}}, \frac{x_{1}^{12} + x_{0}^{10} + 2 x_{0}^{8} x_{1}^{2} + 3 x_{0}^{6} x_{1}^{4} + 4 x_{0}^{4} x_{1}^{6} + 5 x_{0}^{2} x_{1}^{8} + 6 x_{1}^{10} + 5 x_{0}^{8} + 12 x_{0}^{6} x_{1}^{2} + 18 x_{0}^{4} x_{1}^{4} + 20 x_{0}^{2} x_{1}^{6} + 15 x_{1}^{8} + 10 x_{0}^{6} + 24 x_{0}^{4} x_{1}^{2} + 30 x_{0}^{2} x_{1}^{4} + 20 x_{1}^{6} + 10 x_{0}^{4} + 20 x_{0}^{2} x_{1}^{2} + 15 x_{1}^{4} + 5 x_{0}^{2} + 6 x_{1}^{2} + 1}{x_{0}^{6} x_{1}^{5}}\right] }}} {{{id=12| S2.variable_class() /// Traceback (most recent call last): File "", line 1, in File "_sage_input_80.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UzIudmFyaWFibGVfY2xhc3MoKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in File "/private/tmp/tmpZb6MEX/___code___.py", line 2, in exec compile(u'S2.variable_class()' + '\n', '', 'single') File "", line 1, in File "/Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py", line 1455, in variable_class assert self.is_finite(), 'The variable class can - for infinite types - only be computed up to a given depth' AssertionError: The variable class can - for infinite types - only be computed up to a given depth }}} {{{id=14| S2.variable_class(depth=3) /// Found a bipartite seed - restarting the depth counter at zero and constructing the variable class using its bipartite belt. \newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}, \frac{x_{1}^{2} + 1}{x_{0}}, \frac{x_{1}^{4} + x_{0}^{2} + 2 x_{1}^{2} + 1}{x_{0}^{2} x_{1}}, \frac{x_{1}^{6} + x_{0}^{4} + 2 x_{0}^{2} x_{1}^{2} + 3 x_{1}^{4} + 2 x_{0}^{2} + 3 x_{1}^{2} + 1}{x_{0}^{3} x_{1}^{2}}, \frac{x_{1}^{8} + x_{0}^{6} + 2 x_{0}^{4} x_{1}^{2} + 3 x_{0}^{2} x_{1}^{4} + 4 x_{1}^{6} + 3 x_{0}^{4} + 6 x_{0}^{2} x_{1}^{2} + 6 x_{1}^{4} + 3 x_{0}^{2} + 4 x_{1}^{2} + 1}{x_{0}^{4} x_{1}^{3}}, \frac{x_{1}^{10} + x_{0}^{8} + 2 x_{0}^{6} x_{1}^{2} + 3 x_{0}^{4} x_{1}^{4} + 4 x_{0}^{2} x_{1}^{6} + 5 x_{1}^{8} + 4 x_{0}^{6} + 9 x_{0}^{4} x_{1}^{2} + 12 x_{0}^{2} x_{1}^{4} + 10 x_{1}^{6} + 6 x_{0}^{4} + 12 x_{0}^{2} x_{1}^{2} + 10 x_{1}^{4} + 4 x_{0}^{2} + 5 x_{1}^{2} + 1}{x_{0}^{5} x_{1}^{4}}, \frac{x_{1}^{12} + x_{0}^{10} + 2 x_{0}^{8} x_{1}^{2} + 3 x_{0}^{6} x_{1}^{4} + 4 x_{0}^{4} x_{1}^{6} + 5 x_{0}^{2} x_{1}^{8} + 6 x_{1}^{10} + 5 x_{0}^{8} + 12 x_{0}^{6} x_{1}^{2} + 18 x_{0}^{4} x_{1}^{4} + 20 x_{0}^{2} x_{1}^{6} + 15 x_{1}^{8} + 10 x_{0}^{6} + 24 x_{0}^{4} x_{1}^{2} + 30 x_{0}^{2} x_{1}^{4} + 20 x_{1}^{6} + 10 x_{0}^{4} + 20 x_{0}^{2} x_{1}^{2} + 15 x_{1}^{4} + 5 x_{0}^{2} + 6 x_{1}^{2} + 1}{x_{0}^{6} x_{1}^{5}}, \frac{x_{1}^{14} + x_{0}^{12} + 2 x_{0}^{10} x_{1}^{2} + 3 x_{0}^{8} x_{1}^{4} + 4 x_{0}^{6} x_{1}^{6} + 5 x_{0}^{4} x_{1}^{8} + 6 x_{0}^{2} x_{1}^{10} + 7 x_{1}^{12} + 6 x_{0}^{10} + 15 x_{0}^{8} x_{1}^{2} + 24 x_{0}^{6} x_{1}^{4} + 30 x_{0}^{4} x_{1}^{6} + 30 x_{0}^{2} x_{1}^{8} + 21 x_{1}^{10} + 15 x_{0}^{8} + 40 x_{0}^{6} x_{1}^{2} + 60 x_{0}^{4} x_{1}^{4} + 60 x_{0}^{2} x_{1}^{6} + 35 x_{1}^{8} + 20 x_{0}^{6} + 50 x_{0}^{4} x_{1}^{2} + 60 x_{0}^{2} x_{1}^{4} + 35 x_{1}^{6} + 15 x_{0}^{4} + 30 x_{0}^{2} x_{1}^{2} + 21 x_{1}^{4} + 6 x_{0}^{2} + 7 x_{1}^{2} + 1}{x_{0}^{7} x_{1}^{6}}, \frac{x_{1}^{16} + x_{0}^{14} + 2 x_{0}^{12} x_{1}^{2} + 3 x_{0}^{10} x_{1}^{4} + 4 x_{0}^{8} x_{1}^{6} + 5 x_{0}^{6} x_{1}^{8} + 6 x_{0}^{4} x_{1}^{10} + 7 x_{0}^{2} x_{1}^{12} + 8 x_{1}^{14} + 7 x_{0}^{12} + 18 x_{0}^{10} x_{1}^{2} + 30 x_{0}^{8} x_{1}^{4} + 40 x_{0}^{6} x_{1}^{6} + 45 x_{0}^{4} x_{1}^{8} + 42 x_{0}^{2} x_{1}^{10} + 28 x_{1}^{12} + 21 x_{0}^{10} + 60 x_{0}^{8} x_{1}^{2} + 100 x_{0}^{6} x_{1}^{4} + 120 x_{0}^{4} x_{1}^{6} + 105 x_{0}^{2} x_{1}^{8} + 56 x_{1}^{10} + 35 x_{0}^{8} + 100 x_{0}^{6} x_{1}^{2} + 150 x_{0}^{4} x_{1}^{4} + 140 x_{0}^{2} x_{1}^{6} + 70 x_{1}^{8} + 35 x_{0}^{6} + 90 x_{0}^{4} x_{1}^{2} + 105 x_{0}^{2} x_{1}^{4} + 56 x_{1}^{6} + 21 x_{0}^{4} + 42 x_{0}^{2} x_{1}^{2} + 28 x_{1}^{4} + 7 x_{0}^{2} + 8 x_{1}^{2} + 1}{x_{0}^{8} x_{1}^{7}}, \frac{x_{1}^{18} + x_{0}^{16} + 2 x_{0}^{14} x_{1}^{2} + 3 x_{0}^{12} x_{1}^{4} + 4 x_{0}^{10} x_{1}^{6} + 5 x_{0}^{8} x_{1}^{8} + 6 x_{0}^{6} x_{1}^{10} + 7 x_{0}^{4} x_{1}^{12} + 8 x_{0}^{2} x_{1}^{14} + 9 x_{1}^{16} + 8 x_{0}^{14} + 21 x_{0}^{12} x_{1}^{2} + 36 x_{0}^{10} x_{1}^{4} + 50 x_{0}^{8} x_{1}^{6} + 60 x_{0}^{6} x_{1}^{8} + 63 x_{0}^{4} x_{1}^{10} + 56 x_{0}^{2} x_{1}^{12} + 36 x_{1}^{14} + 28 x_{0}^{12} + 84 x_{0}^{10} x_{1}^{2} + 150 x_{0}^{8} x_{1}^{4} + 200 x_{0}^{6} x_{1}^{6} + 210 x_{0}^{4} x_{1}^{8} + 168 x_{0}^{2} x_{1}^{10} + 84 x_{1}^{12} + 56 x_{0}^{10} + 175 x_{0}^{8} x_{1}^{2} + 300 x_{0}^{6} x_{1}^{4} + 350 x_{0}^{4} x_{1}^{6} + 280 x_{0}^{2} x_{1}^{8} + 126 x_{1}^{10} + 70 x_{0}^{8} + 210 x_{0}^{6} x_{1}^{2} + 315 x_{0}^{4} x_{1}^{4} + 280 x_{0}^{2} x_{1}^{6} + 126 x_{1}^{8} + 56 x_{0}^{6} + 147 x_{0}^{4} x_{1}^{2} + 168 x_{0}^{2} x_{1}^{4} + 84 x_{1}^{6} + 28 x_{0}^{4} + 56 x_{0}^{2} x_{1}^{2} + 36 x_{1}^{4} + 8 x_{0}^{2} + 9 x_{1}^{2} + 1}{x_{0}^{9} x_{1}^{8}}, \frac{x_{1}^{20} + x_{0}^{18} + 2 x_{0}^{16} x_{1}^{2} + 3 x_{0}^{14} x_{1}^{4} + 4 x_{0}^{12} x_{1}^{6} + 5 x_{0}^{10} x_{1}^{8} + 6 x_{0}^{8} x_{1}^{10} + 7 x_{0}^{6} x_{1}^{12} + 8 x_{0}^{4} x_{1}^{14} + 9 x_{0}^{2} x_{1}^{16} + 10 x_{1}^{18} + 9 x_{0}^{16} + 24 x_{0}^{14} x_{1}^{2} + 42 x_{0}^{12} x_{1}^{4} + 60 x_{0}^{10} x_{1}^{6} + 75 x_{0}^{8} x_{1}^{8} + 84 x_{0}^{6} x_{1}^{10} + 84 x_{0}^{4} x_{1}^{12} + 72 x_{0}^{2} x_{1}^{14} + 45 x_{1}^{16} + 36 x_{0}^{14} + 112 x_{0}^{12} x_{1}^{2} + 210 x_{0}^{10} x_{1}^{4} + 300 x_{0}^{8} x_{1}^{6} + 350 x_{0}^{6} x_{1}^{8} + 336 x_{0}^{4} x_{1}^{10} + 252 x_{0}^{2} x_{1}^{12} + 120 x_{1}^{14} + 84 x_{0}^{12} + 280 x_{0}^{10} x_{1}^{2} + 525 x_{0}^{8} x_{1}^{4} + 700 x_{0}^{6} x_{1}^{6} + 700 x_{0}^{4} x_{1}^{8} + 504 x_{0}^{2} x_{1}^{10} + 210 x_{1}^{12} + 126 x_{0}^{10} + 420 x_{0}^{8} x_{1}^{2} + 735 x_{0}^{6} x_{1}^{4} + 840 x_{0}^{4} x_{1}^{6} + 630 x_{0}^{2} x_{1}^{8} + 252 x_{1}^{10} + 126 x_{0}^{8} + 392 x_{0}^{6} x_{1}^{2} + 588 x_{0}^{4} x_{1}^{4} + 504 x_{0}^{2} x_{1}^{6} + 210 x_{1}^{8} + 84 x_{0}^{6} + 224 x_{0}^{4} x_{1}^{2} + 252 x_{0}^{2} x_{1}^{4} + 120 x_{1}^{6} + 36 x_{0}^{4} + 72 x_{0}^{2} x_{1}^{2} + 45 x_{1}^{4} + 9 x_{0}^{2} + 10 x_{1}^{2} + 1}{x_{0}^{10} x_{1}^{9}}, \frac{x_{1}^{22} + x_{0}^{20} + 2 x_{0}^{18} x_{1}^{2} + 3 x_{0}^{16} x_{1}^{4} + 4 x_{0}^{14} x_{1}^{6} + 5 x_{0}^{12} x_{1}^{8} + 6 x_{0}^{10} x_{1}^{10} + 7 x_{0}^{8} x_{1}^{12} + 8 x_{0}^{6} x_{1}^{14} + 9 x_{0}^{4} x_{1}^{16} + 10 x_{0}^{2} x_{1}^{18} + 11 x_{1}^{20} + 10 x_{0}^{18} + 27 x_{0}^{16} x_{1}^{2} + 48 x_{0}^{14} x_{1}^{4} + 70 x_{0}^{12} x_{1}^{6} + 90 x_{0}^{10} x_{1}^{8} + 105 x_{0}^{8} x_{1}^{10} + 112 x_{0}^{6} x_{1}^{12} + 108 x_{0}^{4} x_{1}^{14} + 90 x_{0}^{2} x_{1}^{16} + 55 x_{1}^{18} + 45 x_{0}^{16} + 144 x_{0}^{14} x_{1}^{2} + 280 x_{0}^{12} x_{1}^{4} + 420 x_{0}^{10} x_{1}^{6} + 525 x_{0}^{8} x_{1}^{8} + 560 x_{0}^{6} x_{1}^{10} + 504 x_{0}^{4} x_{1}^{12} + 360 x_{0}^{2} x_{1}^{14} + 165 x_{1}^{16} + 120 x_{0}^{14} + 420 x_{0}^{12} x_{1}^{2} + 840 x_{0}^{10} x_{1}^{4} + 1225 x_{0}^{8} x_{1}^{6} + 1400 x_{0}^{6} x_{1}^{8} + 1260 x_{0}^{4} x_{1}^{10} + 840 x_{0}^{2} x_{1}^{12} + 330 x_{1}^{14} + 210 x_{0}^{12} + 756 x_{0}^{10} x_{1}^{2} + 1470 x_{0}^{8} x_{1}^{4} + 1960 x_{0}^{6} x_{1}^{6} + 1890 x_{0}^{4} x_{1}^{8} + 1260 x_{0}^{2} x_{1}^{10} + 462 x_{1}^{12} + 252 x_{0}^{10} + 882 x_{0}^{8} x_{1}^{2} + 1568 x_{0}^{6} x_{1}^{4} + 1764 x_{0}^{4} x_{1}^{6} + 1260 x_{0}^{2} x_{1}^{8} + 462 x_{1}^{10} + 210 x_{0}^{8} + 672 x_{0}^{6} x_{1}^{2} + 1008 x_{0}^{4} x_{1}^{4} + 840 x_{0}^{2} x_{1}^{6} + 330 x_{1}^{8} + 120 x_{0}^{6} + 324 x_{0}^{4} x_{1}^{2} + 360 x_{0}^{2} x_{1}^{4} + 165 x_{1}^{6} + 45 x_{0}^{4} + 90 x_{0}^{2} x_{1}^{2} + 55 x_{1}^{4} + 10 x_{0}^{2} + 11 x_{1}^{2} + 1}{x_{0}^{11} x_{1}^{10}}, \frac{x_{1}^{24} + x_{0}^{22} + 2 x_{0}^{20} x_{1}^{2} + 3 x_{0}^{18} x_{1}^{4} + 4 x_{0}^{16} x_{1}^{6} + 5 x_{0}^{14} x_{1}^{8} + 6 x_{0}^{12} x_{1}^{10} + 7 x_{0}^{10} x_{1}^{12} + 8 x_{0}^{8} x_{1}^{14} + 9 x_{0}^{6} x_{1}^{16} + 10 x_{0}^{4} x_{1}^{18} + 11 x_{0}^{2} x_{1}^{20} + 12 x_{1}^{22} + 11 x_{0}^{20} + 30 x_{0}^{18} x_{1}^{2} + 54 x_{0}^{16} x_{1}^{4} + 80 x_{0}^{14} x_{1}^{6} + 105 x_{0}^{12} x_{1}^{8} + 126 x_{0}^{10} x_{1}^{10} + 140 x_{0}^{8} x_{1}^{12} + 144 x_{0}^{6} x_{1}^{14} + 135 x_{0}^{4} x_{1}^{16} + 110 x_{0}^{2} x_{1}^{18} + 66 x_{1}^{20} + 55 x_{0}^{18} + 180 x_{0}^{16} x_{1}^{2} + 360 x_{0}^{14} x_{1}^{4} + 560 x_{0}^{12} x_{1}^{6} + 735 x_{0}^{10} x_{1}^{8} + 840 x_{0}^{8} x_{1}^{10} + 840 x_{0}^{6} x_{1}^{12} + 720 x_{0}^{4} x_{1}^{14} + 495 x_{0}^{2} x_{1}^{16} + 220 x_{1}^{18} + 165 x_{0}^{16} + 600 x_{0}^{14} x_{1}^{2} + 1260 x_{0}^{12} x_{1}^{4} + 1960 x_{0}^{10} x_{1}^{6} + 2450 x_{0}^{8} x_{1}^{8} + 2520 x_{0}^{6} x_{1}^{10} + 2100 x_{0}^{4} x_{1}^{12} + 1320 x_{0}^{2} x_{1}^{14} + 495 x_{1}^{16} + 330 x_{0}^{14} + 1260 x_{0}^{12} x_{1}^{2} + 2646 x_{0}^{10} x_{1}^{4} + 3920 x_{0}^{8} x_{1}^{6} + 4410 x_{0}^{6} x_{1}^{8} + 3780 x_{0}^{4} x_{1}^{10} + 2310 x_{0}^{2} x_{1}^{12} + 792 x_{1}^{14} + 462 x_{0}^{12} + 1764 x_{0}^{10} x_{1}^{2} + 3528 x_{0}^{8} x_{1}^{4} + 4704 x_{0}^{6} x_{1}^{6} + 4410 x_{0}^{4} x_{1}^{8} + 2772 x_{0}^{2} x_{1}^{10} + 924 x_{1}^{12} + 462 x_{0}^{10} + 1680 x_{0}^{8} x_{1}^{2} + 3024 x_{0}^{6} x_{1}^{4} + 3360 x_{0}^{4} x_{1}^{6} + 2310 x_{0}^{2} x_{1}^{8} + 792 x_{1}^{10} + 330 x_{0}^{8} + 1080 x_{0}^{6} x_{1}^{2} + 1620 x_{0}^{4} x_{1}^{4} + 1320 x_{0}^{2} x_{1}^{6} + 495 x_{1}^{8} + 165 x_{0}^{6} + 450 x_{0}^{4} x_{1}^{2} + 495 x_{0}^{2} x_{1}^{4} + 220 x_{1}^{6} + 55 x_{0}^{4} + 110 x_{0}^{2} x_{1}^{2} + 66 x_{1}^{4} + 11 x_{0}^{2} + 12 x_{1}^{2} + 1}{x_{0}^{12} x_{1}^{11}}\right] }}} {{{id=18| S2.b_matrix_class(); S2.b_matrix_class(up_to_equivalence=False) /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right)\right] \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right), \left(\begin{array}{rr} 0 & -2 \\ 2 & 0 \end{array}\right)\right] }}} {{{id=20| VV = S2.variable_class(depth=3); DD = map(denominator,VV); DD /// Found a bipartite seed - restarting the depth counter at zero and constructing the variable class using its bipartite belt. \newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, x_{0}, x_{0}^{2} x_{1}, x_{0}^{3} x_{1}^{2}, x_{0}^{4} x_{1}^{3}, x_{0}^{5} x_{1}^{4}, x_{0}^{6} x_{1}^{5}, x_{0}^{7} x_{1}^{6}, x_{0}^{8} x_{1}^{7}, x_{0}^{9} x_{1}^{8}, x_{0}^{10} x_{1}^{9}, x_{0}^{11} x_{1}^{10}, x_{0}^{12} x_{1}^{11}\right] }}} {{{id=21| [monom.degrees() for monom in DD] /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(0, 0\right), \left(0, 0\right), \left(1, 0\right), \left(2, 1\right), \left(3, 2\right), \left(4, 3\right), \left(5, 4\right), \left(6, 5\right), \left(7, 6\right), \left(8, 7\right), \left(9, 8\right), \left(10, 9\right), \left(11, 10\right), \left(12, 11\right)\right] }}} {{{id=133| /// }}} {{{id=132| S33 = ClusterSeed(['A',[3,3],1]); S33 /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 6 of type ['A', [3, 3], 1]} }}} {{{id=134| S33.show() /// }}} {{{id=39| MC = S33.b_matrix_class(); MC /// WARNING: Output truncated! full_output.txt \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 & 0 \\ -1 & -1 & 0 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 1 \\ -1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 1 \\ 0 & -1 & 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & 1 & 1 \\ -1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & -1 & 0 \\ 0 & -1 & -1 & 1 & 0 & 0 \\ -1 & -1 & 0 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 & 0 & 1 \\ -1 & 0 & 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & -1 & 0 & 0 & 1 \\ -1 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 1 & 1 \\ 0 & 0 & -1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 & 0 & -1 \\ -1 & -1 & 0 & 0 & 0 & 0 \\ -1 & -1 & 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & 1 & 0 \\ ... 0 & 1 & 1 & 0 & -1 & -1 \\ -1 & 0 & -1 & 1 & 0 & 0 \\ -1 & -1 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 \\ 1 & 0 & 0 & 1 & 0 & -1 \\ 0 & -1 & 0 & -1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 1 & 0 & 0 & 0 & 1 & -1 \\ 0 & -1 & 1 & -1 & 0 & 1 \\ 0 & 0 & -1 & 1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & -1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & -1 \\ 0 & -1 & -1 & 0 & 0 & 1 \\ 0 & 1 & -1 & 1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 1 & -1 \\ 0 & -1 & 0 & -1 & 0 & 1 \\ -1 & 0 & 0 & 1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 & 1 & 1 \\ -1 & 0 & 1 & 0 & -1 & 0 \\ 0 & 1 & -1 & 1 & 0 & -1 \\ 0 & -1 & -1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 1 \\ -1 & 1 & 0 & 0 & 0 & 1 \\ 2 & 0 & 0 & -1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & -2 \\ 0 & -1 & -1 & 0 & 0 & 1 \\ -1 & 0 & -1 & 0 & 0 & 1 \\ 0 & 0 & 2 & -1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & -2 \\ 0 & -1 & 0 & 1 & 0 & 1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & -1 & 0 & 0 & 0 & 1 \\ 0 & 2 & -1 & 0 & -1 & 0 \end{array}\right)\right] }}} {{{id=40| B = MC[5]; B /// \newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 \\ 0 & 1 & -1 & 0 & 0 & 1 \\ -1 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 & 0 & 0 \end{array}\right) }}} {{{id=41| Snew = ClusterSeed(B); Snew; Snew.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 6} }}} {{{id=42| Snew2 = Snew.principal_extension(); Snew2; Snew2.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 6 with 6 frozen variables} }}} {{{id=43| Snew2.mutation_type() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{['A', [3, 3], 1]} }}} {{{id=44| Snew2 /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 6 of type ['A', [3, 3], 1] with 6 frozen variables} }}} {{{id=45| QuiverMutationType? ///

File: /Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/quiver_mutation_type.py

Type: <class ‘sage.combinat.cluster_algebra_quiver.quiver_mutation_type.QuiverMutationTypeFactory’>

Definition: QuiverMutationType(*args)

Docstring:

Quiver mutation types can be seen as a slight generalization of generalized Cartan types.

Background on generalized Cartan types can be found at

http://en.wikipedia.org/wiki/Generalized_Cartan_matrix

For the compendium on the cluster algebra and quiver package in Sage see

http://arxiv.org/abs/1102.4844

A B-matrix is a skew-symmetrizable ( n x n )-matrix M. I.e., there exists an invertible diagonal matrix D such that DM is skew-symmetric. M can be encoded as a quiver by having a directed edge from vertex i to vertex j with label (a,b) if a = M_{i,j} > 0 and b = M_{j,i} < 0. We consider quivers up to mutation equivalence.

In particular, to a quiver mutation type we can associate a generalized Cartan type by sending M to the generalized Cartan matrix C(M) obtained by replacing all positive entries by their negatives and adding 2‘s on the main diagonal.

It appears that C(M) and C(M') are isomorphic Cartan types for mutation equivalent skew-symmetrizable matrices M and M'. Thus, all generalized Cartan types appear as well as quiver mutation types.

AUTHOR:
– Gregg Musiker – Christian Stump

Constructs a quiver mutation type object. For the possible different types, please see the compendium. Kac’s classification types can also be used as input.

INPUT:

  • letter, rank – letter is one of ‘A’,’B’,’C’,’D’,’E’,’F’,’G’ and rank is an integer
  • letter, rank, twist – letter is one of ‘A’,’BB’,’CC’,’D’,’E’,’F’,’G’, ‘BC’, ‘BD’, ‘CD’, and rank is a tuple (b,c) or an integer and twist is an integer
  • object – a quiver mutation type

EXAMPLES:

Finite types:

sage: QuiverMutationType('A',1)
['A', 1]
sage: QuiverMutationType('A',5)
['A', 5]

sage: QuiverMutationType('B',2)
['B', 2]
sage: QuiverMutationType('B',5)
['B', 5]

sage: QuiverMutationType('C',2)
['B', 2]
sage: QuiverMutationType('C',5)
['C', 5]

sage: QuiverMutationType('D',2)
[ ['A', 1], ['A', 1] ]
sage: QuiverMutationType('D',3)
['A', 3]
sage: QuiverMutationType('D',4)
['D', 4]

sage: QuiverMutationType('E',6)
['E', 6]
sage: QuiverMutationType('E',7)
['E', 7]
sage: QuiverMutationType('E',8)
['E', 8]

sage: QuiverMutationType('F',4)
['F', 4]

sage: QuiverMutationType('G',2)
['G', 2]

Affine types:

sage: QuiverMutationType('A',(1,1),1)
['A', [1, 1], 1]
sage: QuiverMutationType('A',(2,4),1)
['A', [2, 4], 1]

sage: QuiverMutationType('BB',1,1)
['A', [1, 1], 1]
sage: QuiverMutationType('BB',2,1)
['BB', 2, 1]
sage: QuiverMutationType('BB',4,1)
['BB', 4, 1]

sage: QuiverMutationType('CC',1,1)
['A', [1, 1], 1]
sage: QuiverMutationType('CC',2,1)
['CC', 2, 1]
sage: QuiverMutationType('CC',4,1)
['CC', 4, 1]

sage: QuiverMutationType('BC',1,1)
['BC', 1, 1]
sage: QuiverMutationType('BC',5,1)
['BC', 5, 1]

sage: QuiverMutationType('BD',3,1)
['BD', 3, 1]
sage: QuiverMutationType('BD',5,1)
['BD', 5, 1]

sage: QuiverMutationType('CD',3,1)
['CD', 3, 1]
sage: QuiverMutationType('CD',5,1)
['CD', 5, 1]

sage: QuiverMutationType('D',4,1)
['D', 4, 1]
sage: QuiverMutationType('D',6,1)
['D', 6, 1]

sage: QuiverMutationType('E',6,1)
['E', 6, 1]
sage: QuiverMutationType('E',7,1)
['E', 7, 1]
sage: QuiverMutationType('E',8,1)
['E', 8, 1]

sage: QuiverMutationType('F',4,1)
['F', 4, 1]
sage: QuiverMutationType('F',4,-1)
['F', 4, -1]

sage: QuiverMutationType('G',2,1)
['G', 2, 1]
sage: QuiverMutationType('G',2,-1)
['G', 2, -1]

Elliptic types:

sage: QuiverMutationType('E',6,[1,1])
['E', 6, [1, 1]]
sage: QuiverMutationType('E',7,[1,1])
['E', 7, [1, 1]]
sage: QuiverMutationType('E',8,[1,1])
['E', 8, [1, 1]]

Mutation finite types:

rank 2 cases:

sage: QuiverMutationType('R2',(1,1),2)
['A', 2]
sage: QuiverMutationType('R2',(1,2),2)
['B', 2]
sage: QuiverMutationType('R2',(1,3),2)
['G', 2]
sage: QuiverMutationType('R2',(1,4),2)
['BC', 1, 1]
sage: QuiverMutationType('R2',(1,5),2)
['R2', [1, 5], 2]
sage: QuiverMutationType('R2',(2,2),2)
['A', [1, 1], 1]
sage: QuiverMutationType('R2',(3,5),2)
['R2', [3, 5], 2]

exceptional quiver mutation types:

sage: QuiverMutationType('V',4,2)
['V', 4, 2]
sage: QuiverMutationType('W',4,2)
['W', 4, 2]
sage: QuiverMutationType('W',4,-2)
['W', 4, -2]
sage: QuiverMutationType('X',6,2)
['X', 6, 2]
sage: QuiverMutationType('Y',6,2)
['Y', 6, 2]
sage: QuiverMutationType('Z',6,2)
['Z', 6, 2]
sage: QuiverMutationType('Z',6,-2)
['Z', 6, -2]

Mutation infinite types:

infinite type E:

sage: QuiverMutationType('E',9,3)
['E', 8, 1]
sage: QuiverMutationType('E',10,3)
['E', 10, 3]
sage: QuiverMutationType('E',12,3)
['E', 12, 3]

sage: QuiverMutationType('AE',(1,1),3)
['AE', [1, 1], 3]
sage: QuiverMutationType('AE',(1,4),3)
['AE', [1, 4], 3]
sage: QuiverMutationType('BE',5,3)
['BE', 5, 3]
sage: QuiverMutationType('CE',5,3)
['CE', 5, 3]
sage: QuiverMutationType('DE',6,3)
['DE', 6, 3]

Grassmannian types:

sage: QuiverMutationType('GR',(2,4),3)
['A', 1]
sage: QuiverMutationType('GR',(2,6),3)
['A', 3]
sage: QuiverMutationType('GR',(3,6),3)
['D', 4]
sage: QuiverMutationType('GR',(3,7),3)
['E', 6]
sage: QuiverMutationType('GR',(3,8),3)
['E', 8]
sage: QuiverMutationType('GR',(3,10),3)
['GR', [3, 10], 3]

Triangular types:

sage: QuiverMutationType('TR',2,3)
['A', 3]
sage: QuiverMutationType('TR',3,3)
['D', 6]
sage: QuiverMutationType('TR',4,3)
['E', 8, [1, 1]]
sage: QuiverMutationType('TR',5,3)
['TR', 5, 3]

T types:

sage: QuiverMutationType('T',(1,1,1),3)
['A', 1]
sage: QuiverMutationType('T',(1,1,4),3)
['A', 4]
sage: QuiverMutationType('T',(1,4,4),3)
['A', 7]
sage: QuiverMutationType('T',(2,2,2),3)
['D', 4]
sage: QuiverMutationType('T',(2,2,4),3)
['D', 6]
sage: QuiverMutationType('T',(2,3,3),3)
['E', 6]
sage: QuiverMutationType('T',(2,3,4),3)
['E', 7]
sage: QuiverMutationType('T',(2,3,5),3)
['E', 8]
sage: QuiverMutationType('T',(2,3,6),3)
['E', 8, 1]
sage: QuiverMutationType('T',(2,3,7),3)
['E', 10, 3]
sage: QuiverMutationType('T',(3,3,3),3)
['E', 6, 1]
sage: QuiverMutationType('T',(3,3,4),3)
['T', [3, 3, 4], 3]

Reducible types:

sage: QuiverMutationType(['A',3],['B',4])
[ ['A', 3], ['B', 4] ]
}}} {{{id=135| Gr = ClusterSeed(['GR',[4,9],3]); Gr; Gr.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 12 of type ['GR', [4, 9], 3]} }}} {{{id=136| Gr.is_mutation_finite() /// \newcommand{\Bold}[1]{\mathbf{#1}}{\rm False} }}} {{{id=137| Gr2 = ClusterSeed(['GR',[4,8],3]); Gr2; /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 9 of type ['E', 7, [1, 1]]} }}} {{{id=139| Gr2.is_mutation_finite() /// \newcommand{\Bold}[1]{\mathbf{#1}}{\rm True} }}} {{{id=140| Tr = ClusterSeed(['TR',5,3]); Tr; Tr.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 15 of type ['TR', 5, 3]} }}} {{{id=138| NSL = ClusterSeed(['F',4,1]); NSL; NSL.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 5 of type ['F', 4, 1]} }}} {{{id=141| NSL2 = ClusterSeed(['F',4,-1]); NSL2; NSL2.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 5 of type ['F', 4, -1]} }}} {{{id=142| BB = NSL.b_matrix_class(); len(BB); /// \newcommand{\Bold}[1]{\mathbf{#1}}60 }}} {{{id=144| BB /// WARNING: Output truncated! full_output.txt \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & -1 & -2 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & -2 & 0 & 1 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ -1 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 1 & 0 \\ 0 & -1 & 0 & 0 & 1 \\ 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 2 & 0 \\ 0 & 0 & -1 & 0 & -1 \\ 0 & -1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 1 \\ -1 & 0 & 0 & -2 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & -1 \\ -1 & -2 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & -2 & 0 & 1 & 0 \\ 0 & 2 & -1 & 0 & 0 \\ -1 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 1 \\ -2 & 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 \\ ... 0 & 0 & -2 & 1 & 0 \\ 0 & 0 & 0 & -1 & 1 \\ 1 & 0 & 0 & -1 & 1 \\ -1 & 2 & 2 & 0 & -2 \\ 0 & -1 & -1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & -1 & 0 & 1 \\ 0 & 0 & 1 & -2 & 0 \\ 2 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 2 & 1 & -2 & -2 \\ -1 & 0 & 0 & 1 & 1 \\ -1 & 0 & 0 & 0 & 2 \\ 1 & -1 & 0 & 0 & 0 \\ 1 & -1 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 0 & 2 & -2 \\ -1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & -1 \\ -1 & 0 & -1 & 0 & 1 \\ 1 & -1 & 1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 0 & -1 & 1 \\ -2 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & -1 \\ 2 & -1 & -2 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & -1 & 0 & 0 & 1 \\ 2 & 0 & 1 & -2 & 0 \\ 0 & -1 & 0 & 2 & 0 \\ 0 & 1 & -1 & 0 & -1 \\ -1 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & -1 & 0 & 0 & 1 \\ -1 & 2 & 0 & 0 & -2 \\ 0 & -1 & -1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & -1 & 0 & 1 & 1 \\ 2 & 0 & 1 & 0 & -2 \\ 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 1 \\ -1 & 1 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 & 1 \\ 0 & 2 & 0 & -1 & -1 \\ -1 & -2 & 2 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 0 & -2 & 1 \\ -2 & 0 & 1 & 2 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 2 & -1 & 0 & 0 & -1 \\ -1 & 0 & 0 & 1 & 0 \end{array}\right)\right] }}} {{{id=143| BB2 = NSL2.b_matrix_class(); len(BB2) /// \newcommand{\Bold}[1]{\mathbf{#1}}60 }}} {{{id=145| BB2 = NSL2.b_matrix_class(up_to_equivalence=False); len(BB2) /// \newcommand{\Bold}[1]{\mathbf{#1}}720 }}} {{{id=146| for Mat in BB: if Mat in BB2: print("Found Matrix") print("Done") /// Done }}} {{{id=49| NSL.interact() /// }}} {{{id=50| EE7 = ClusterSeed(['E',7]); EE7.show() /// }}} {{{id=99| VC = EE7.variable_class(); len(VC) /// \newcommand{\Bold}[1]{\mathbf{#1}}70 }}} {{{id=54| VC[35] /// \newcommand{\Bold}[1]{\mathbf{#1}}\frac{x_{0} x_{2}^{2} x_{4}^{2} + x_{1} x_{3}^{2} x_{5} x_{6} + x_{0} x_{2}^{2} x_{4} + x_{0} x_{2} x_{3} x_{5} + x_{1} x_{3} x_{4} x_{6} + x_{0} x_{2} x_{4} + x_{2} x_{4}^{2} + x_{1} x_{3} x_{6} + x_{0} x_{2} + x_{2} x_{4} + x_{3} x_{5} + x_{4} + 1}{x_{1} x_{2} x_{3} x_{4} x_{5}} }}} {{{id=60| for i in range(len(VC)): if max(VC[i].denominator().exponents()[0]) > 1: print(i) /// 34 38 41 43 44 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 }}} {{{id=61| VC[34] /// \newcommand{\Bold}[1]{\mathbf{#1}}\frac{x_{1}^{2} x_{3}^{2} x_{6}^{2} + x_{0} x_{2}^{3} x_{4} + x_{0} x_{1} x_{2} x_{3} x_{6} + x_{1} x_{2} x_{3} x_{4} x_{6} + x_{0} x_{2}^{2} x_{4} + x_{1} x_{2} x_{3} x_{6} + x_{0} x_{2}^{2} + x_{2}^{2} x_{4} + 2 x_{1} x_{3} x_{6} + x_{0} x_{2} + x_{2} x_{4} + x_{2} + 1}{x_{1} x_{2}^{2} x_{3} x_{6}} }}} {{{id=67| X = ClusterSeed(['X',6,2]); X; X.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 6 of type ['X', 6, 2]} }}} {{{id=68| S = ClusterSeed(['X',7,2]); S; S.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 7 of type ['X', 7, 2]} }}} {{{id=71| X.is_finite() /// \newcommand{\Bold}[1]{\mathbf{#1}}{\rm False} }}} {{{id=69| X.is_mutation_finite() /// \newcommand{\Bold}[1]{\mathbf{#1}}{\rm True} }}} {{{id=70| V = ClusterSeed(['V',4,2]); V; V.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 4 of type ['V', 4, 2]} }}} {{{id=72| VMC = V.b_matrix_class(); len(VMC); VMC /// \newcommand{\Bold}[1]{\mathbf{#1}}7 \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrr} 0 & 0 & -1 & 1 \\ 0 & 0 & 1 & -3 \\ 3 & -1 & 0 & 0 \\ -1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & -3 & 0 & 1 \\ 1 & 0 & 1 & -1 \\ 0 & -1 & 0 & 1 \\ -1 & 3 & -3 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & -1 & 0 & 1 \\ 3 & 0 & 1 & -3 \\ 0 & -1 & 0 & 3 \\ -1 & 1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & -1 & 0 \\ -2 & 0 & 1 & 0 \\ 3 & -3 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & -2 & 3 \\ 0 & 2 & 0 & -3 \\ -1 & -1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 3 & -2 & 0 \\ -1 & 0 & 1 & 1 \\ 2 & -3 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & -1 \\ 0 & -2 & 0 & 1 \\ -1 & 3 & -3 & 0 \end{array}\right)\right] }}} {{{id=81| WW = ClusterSeed(['W',4,2]); WW; WW.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 4 of type ['W', 4, 2]} }}} {{{id=147| WW2 = ClusterSeed(['W',4,-2]); WW2; WW2.show() /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 4 of type ['W', 4, -2]} }}} {{{id=148| WW.b_matrix_class() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrr} 0 & 1 & 1 & -2 \\ -3 & 0 & 0 & 3 \\ -1 & 0 & 0 & 1 \\ 2 & -1 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & -1 & 1 & 1 \\ 3 & 0 & 0 & -3 \\ -1 & 0 & 0 & 1 \\ -1 & 1 & -1 & 0 \end{array}\right)\right] }}} {{{id=149| WW2.b_matrix_class() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrrr} 0 & 3 & 1 & -2 \\ -1 & 0 & 0 & 1 \\ -1 & 0 & 0 & 1 \\ 2 & -3 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & -1 \\ -3 & 0 & 1 & 1 \\ 0 & -1 & 0 & 1 \\ 3 & -1 & -1 & 0 \end{array}\right)\right] }}} {{{id=97| So4 = Matrix([[0,-1,2,-1],[1,0,-3,2],[-2,3,0,-1],[1,-2,1,0]]) /// }}} {{{id=150| Somos4 = ClusterSeed(So4); Somos4 /// \newcommand{\Bold}[1]{\mathbf{#1}}\hbox{A seed for a cluster algebra of rank 4} }}} {{{id=95| Somos4.set_cluster([1,1,1,1]) /// }}} {{{id=77| Somos4.show() /// }}} {{{id=151| Somos4.mutate([0,1,2,3]); Somos4.show(); Somos4.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[2, 3, 7, 23\right] }}} {{{id=152| Somos4.mutate([0,1,2,3]); Somos4.show(); Somos4.cluster() /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[59, 314, 1529, 8209\right] }}} {{{id=153| Somos4.mutation_sequence([0,1,2,3,0,1,2,3],return_output='var') /// \newcommand{\Bold}[1]{\mathbf{#1}}\left[83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786\right] }}} {{{id=154| /// }}}