Cluster Package demo
system:sage


{{{id=158|
B = Matrix([[0,1],[-1,0]])
///
}}}

{{{id=101|
B
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}\right)</span></html>
}}}

{{{id=102|
ClusterSeed?
///
<html><!--notruncate-->

<div class="docstring">
    
  <p><strong>File:</strong> /Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py</p>
<p><strong>Type:</strong> &lt;type &#8216;type&#8217;&gt;</p>
<p><strong>Definition:</strong> ClusterSeed( [noargspec] )</p>
<p><strong>Docstring:</strong></p>
<blockquote>
<p>The <em>cluster seed</em> associated to an <em>exchange matrix</em>.</p>
<p>INPUT:</p>
<ul>
<li><p class="first"><tt class="docutils literal"><span class="pre">data</span></tt> &#8211; can be any of the following:</p>
<div class="highlight-python"><pre class="literal-block">* QuiverMutationType
* str - a string representing a QuiverMutationType
* Quiver
* 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</pre>
</div>
</li>
</ul>
<p>EXAMPLES:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">S</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">([</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">]);</span> <span class="n">S</span>
<span class="go">A seed for a cluster algebra of rank 5 of type [&#39;A&#39;, 5]</span>

<span class="gp">sage: </span><span class="n">S</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">([</span><span class="s">&#39;A&#39;</span><span class="p">,[</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">],</span><span class="mi">1</span><span class="p">]);</span> <span class="n">S</span>
<span class="go">A seed for a cluster algebra of rank 7 of type [&#39;A&#39;, [2, 5], 1]</span>

<span class="gp">sage: </span><span class="n">T</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">(</span> <span class="n">S</span> <span class="p">);</span> <span class="n">T</span>
<span class="go">A seed for a cluster algebra of rank 7 of type [&#39;A&#39;, [2, 5], 1]</span>

<span class="gp">sage: </span><span class="n">T</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">(</span> <span class="n">S</span><span class="o">.</span><span class="n">_M</span> <span class="p">);</span> <span class="n">T</span>
<span class="go">A seed for a cluster algebra of rank 7</span>

<span class="gp">sage: </span><span class="n">T</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">(</span> <span class="n">S</span><span class="o">.</span><span class="n">quiver</span><span class="p">()</span><span class="o">.</span><span class="n">_digraph</span> <span class="p">);</span> <span class="n">T</span>
<span class="go">A seed for a cluster algebra of rank 7</span>

<span class="gp">sage: </span><span class="n">T</span> <span class="o">=</span> <span class="n">ClusterSeed</span><span class="p">(</span> <span class="n">S</span><span class="o">.</span><span class="n">quiver</span><span class="p">()</span><span class="o">.</span><span class="n">_digraph</span><span class="o">.</span><span class="n">edges</span><span class="p">()</span> <span class="p">);</span> <span class="n">T</span>
<span class="go">A seed for a cluster algebra of rank 7</span>
</pre></div>
</div>
</blockquote>


</div>
</html>
}}}

{{{id=103|
S = ClusterSeed(B); S
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|2|</span></html>
}}}

{{{id=104|
S.mutation_type()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|['A',|\phantom{x}\verb|2]|</span></html>
}}}

{{{id=105|
S
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|2|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['A',|\phantom{x}\verb|2]|</span></html>
}}}

{{{id=155|

///
}}}

{{{id=106|
S.is_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=107|
S.is_mutation_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=109|
S.is_acyclic()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=108|
S.is_bipartite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=112|
S.show()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=113|
S.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}\right]</span></html>
}}}

{{{id=111|
S.mutate(0); S.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1} + 1}{x_{0}}, x_{1}\right]</span></html>
}}}

{{{id=114|
S.mutate(1); S.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{1} + 1}{x_{0}}, \frac{x_{0} + x_{1} + 1}{x_{0} x_{1}}\right]</span></html>
}}}

{{{id=116|
S.mutate([0,1]); S.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{0} + 1}{x_{1}}, x_{0}\right]</span></html>
}}}

{{{id=117|
S.variable_class()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=156|
S.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{x_{0} + 1}{x_{1}}, x_{0}\right]</span></html>
}}}

{{{id=115|
S.reset_cluster()
///
}}}

{{{id=118|
SP = S.principal_extension(); SP
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|2|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['A',|\phantom{x}\verb|2]|\phantom{x}\verb|with|\phantom{x}\verb|2|\phantom{x}\verb|frozen|\phantom{x}\verb|variables|</span></html>
}}}

{{{id=161|
SP.b_matrix()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
0 & 1 \\
-1 & 0 \\
1 & 0 \\
0 & 1
\end{array}\right)</span></html>
}}}

{{{id=119|
SP.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}\right]</span></html>
}}}

{{{id=120|
SP.variable_class()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=122|
SP.mutation_sequence([0,1,0,1,0],return_output='matrix')
///
<html><span class="math">\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]</span></html>
}}}

{{{id=123|
Fpolys = SP.variable_class(); Fpolys
///
<html><span class="math">\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]</span></html>
}}}

{{{id=125|
SP.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}\right]</span></html>
}}}

{{{id=124|
SP.set_cluster([1,1,SP.y(0),SP.y(1)]); SP.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1\right]</span></html>
}}}

{{{id=127|

///
}}}

{{{id=1|
S2 = ClusterSeed(['A',[1,1],1]); S2
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|2|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['A',|\phantom{x}\verb|[1,|\phantom{x}\verb|1],|\phantom{x}\verb|1]|</span></html>
}}}

{{{id=110|
S2.b_matrix()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
0 & 2 \\
-2 & 0
\end{array}\right)</span></html>
}}}

{{{id=98|
S2.show()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=3|
S2.cluster()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[x_{0}, x_{1}\right]</span></html>
}}}

{{{id=4|
S2.mutate([0,1]); S2.cluster()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=5|
S2.mutate([0,1]); S2.cluster()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=6|
S2.mutate([0,1]); S2.cluster()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=131|
latex( S2.cluster() )
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|\left[\frac{x_{1}^{10}|\phantom{x}\verb|+|\phantom{x}\verb|x_{0}^{8}|\phantom{x}\verb|+|\phantom{x}\verb|2|\phantom{x}\verb|x_{0}^{6}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|3|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|4|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|5|\phantom{x}\verb|x_{1}^{8}|\phantom{x}\verb|+|\phantom{x}\verb|4|\phantom{x}\verb|x_{0}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|9|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|12|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|10|\phantom{x}\verb|x_{1}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|6|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|12|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|10|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|4|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|5|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|1}{x_{0}^{5}|\phantom{x}\verb|x_{1}^{4}},|\phantom{x}\verb|\frac{x_{1}^{12}|\phantom{x}\verb|+|\phantom{x}\verb|x_{0}^{10}|\phantom{x}\verb|+|\phantom{x}\verb|2|\phantom{x}\verb|x_{0}^{8}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|3|\phantom{x}\verb|x_{0}^{6}|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|4|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|x_{1}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|5|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{8}|\phantom{x}\verb|+|\phantom{x}\verb|6|\phantom{x}\verb|x_{1}^{10}|\phantom{x}\verb|+|\phantom{x}\verb|5|\phantom{x}\verb|x_{0}^{8}|\phantom{x}\verb|+|\phantom{x}\verb|12|\phantom{x}\verb|x_{0}^{6}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|18|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|20|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|15|\phantom{x}\verb|x_{1}^{8}|\phantom{x}\verb|+|\phantom{x}\verb|10|\phantom{x}\verb|x_{0}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|24|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|30|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|20|\phantom{x}\verb|x_{1}^{6}|\phantom{x}\verb|+|\phantom{x}\verb|10|\phantom{x}\verb|x_{0}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|20|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|15|\phantom{x}\verb|x_{1}^{4}|\phantom{x}\verb|+|\phantom{x}\verb|5|\phantom{x}\verb|x_{0}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|6|\phantom{x}\verb|x_{1}^{2}|\phantom{x}\verb|+|\phantom{x}\verb|1}{x_{0}^{6}|\phantom{x}\verb|x_{1}^{5}}\right]|</span></html>
}}}

{{{id=12|
S2.variable_class()
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_39.py", line 10, in <module>
    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 <module>
    
  File "/private/tmp/tmpmgk6BI/___code___.py", line 2, in <module>
    exec compile(u'S2.variable_class()' + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "/Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py", line 1623, 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.
<html><span class="math">\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]</span></html>
}}}

{{{id=18|
S2.b_matrix_class(); S2.b_matrix_class(up_to_equivalence=False)
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rr}
0 & 2 \\
-2 & 0
\end{array}\right)\right]</span></html>
<html><span class="math">\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]</span></html>
}}}

{{{id=163|
VV = S2.variable_class(depth=3); DD = map(denominator,VV)
///
Found a bipartite seed - restarting the depth counter at zero and constructing the variable class using its bipartite belt.
}}}

{{{id=20|
DD
///
<html><span class="math">\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]</span></html>
}}}

{{{id=21|
[monom.degrees() for monom in DD]
///
<html><span class="math">\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]</span></html>
}}}

{{{id=133|

///
}}}

{{{id=132|
S33 = ClusterSeed(['A',[3,3],1]); S33
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|6|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['A',|\phantom{x}\verb|[3,|\phantom{x}\verb|3],|\phantom{x}\verb|1]|</span></html>
}}}

{{{id=134|
S33.show()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=39|
MC = S33.b_matrix_class(); MC
///
WARNING: Output truncated!  
<html><a target='_new' href='/home/stumpc5/8/cells/39/full_output.txt' class='file_link'>full_output.txt</a></html>



<html><span class="math">\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]</span></html>
}}}

{{{id=40|
B = MC[5]; B
///
<html><span class="math">\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)</span></html>
}}}

{{{id=41|
Snew = ClusterSeed(B); Snew; Snew.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|6|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=42|
Snew2 = Snew.principal_extension(); Snew2; Snew2.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|6|\phantom{x}\verb|with|\phantom{x}\verb|6|\phantom{x}\verb|frozen|\phantom{x}\verb|variables|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=43|
Snew2.mutation_type()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|['A',|\phantom{x}\verb|[3,|\phantom{x}\verb|3],|\phantom{x}\verb|1]|</span></html>
}}}

{{{id=44|
Snew2
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|6|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['A',|\phantom{x}\verb|[3,|\phantom{x}\verb|3],|\phantom{x}\verb|1]|\phantom{x}\verb|with|\phantom{x}\verb|6|\phantom{x}\verb|frozen|\phantom{x}\verb|variables|</span></html>
}}}

{{{id=45|
QuiverMutationType?
///
<html><!--notruncate-->

<div class="docstring">
    
  <p><strong>File:</strong> /Users/sage/sage/local/lib/python2.6/site-packages/sage/combinat/cluster_algebra_quiver/quiver_mutation_type.py</p>
<p><strong>Type:</strong> &lt;class &#8216;sage.combinat.cluster_algebra_quiver.quiver_mutation_type.QuiverMutationTypeFactory&#8217;&gt;</p>
<p><strong>Definition:</strong> QuiverMutationType(*args)</p>
<p><strong>Docstring:</strong></p>
<p><em>Quiver mutation types</em> can be seen as a slight generalization of <em>generalized Cartan types</em>.</p>
<p>Background on generalized Cartan types can be found at</p>
<blockquote>
<a class="reference external" href="http://en.wikipedia.org/wiki/Generalized_Cartan_matrix">http://en.wikipedia.org/wiki/Generalized_Cartan_matrix</a></blockquote>
<p>For the compendium on the cluster algebra and quiver package in Sage see</p>
<blockquote>
<a class="reference external" href="http://arxiv.org/abs/1102.4844">http://arxiv.org/abs/1102.4844</a></blockquote>
<p>A <span class="math">B</span>-matrix is a skew-symmetrizable <span class="math">( n x n )</span>-matrix <span class="math">M</span>.
I.e., there exists an invertible diagonal matrix <span class="math">D</span> such that <span class="math">DM</span> is skew-symmetric.
<span class="math">M</span> can be encoded as a <em>quiver</em> by having a directed edge from vertex <span class="math">i</span> to vertex <span class="math">j</span>
with label <span class="math">(a,b)</span> if <span class="math">a = M_{i,j} &gt; 0</span> and <span class="math">b = M_{j,i} &lt; 0</span>.
We consider quivers up to <em>mutation equivalence</em>.</p>
<p>In particular, to a quiver mutation type we can associate a <em>generalized Cartan type</em>
by sending <span class="math">M</span> to the generalized Cartan matrix <span class="math">C(M)</span> obtained by replacing all
positive entries by their negatives and adding <span class="math">2</span>&#8216;s on the main diagonal.</p>
<p>It appears that <span class="math">C(M)</span> and <span class="math">C(M')</span> are isomorphic Cartan types for
mutation equivalent skew-symmetrizable matrices <span class="math">M</span> and <span class="math">M'</span>.
Thus, all generalized Cartan types appear as well as quiver mutation types.</p>
<dl class="docutils">
<dt>AUTHOR:</dt>
<dd>&#8211; Gregg Musiker
&#8211; Christian Stump</dd>
</dl>
<p>Constructs a quiver mutation type object. For the possible different types, please see the compendium.
Kac&#8217;s classification types can also be used as input.</p>
<p>INPUT:</p>
<blockquote>
<ul class="simple">
<li><tt class="docutils literal"><span class="pre">letter,</span> <span class="pre">rank</span></tt> &#8211; letter is one of &#8216;A&#8217;,&#8217;B&#8217;,&#8217;C&#8217;,&#8217;D&#8217;,&#8217;E&#8217;,&#8217;F&#8217;,&#8217;G&#8217; and rank is an integer</li>
<li><tt class="docutils literal"><span class="pre">letter,</span> <span class="pre">rank,</span> <span class="pre">twist</span></tt> &#8211; letter is one of &#8216;A&#8217;,&#8217;BB&#8217;,&#8217;CC&#8217;,&#8217;D&#8217;,&#8217;E&#8217;,&#8217;F&#8217;,&#8217;G&#8217;, &#8216;BC&#8217;, &#8216;BD&#8217;, &#8216;CD&#8217;, and rank is a tuple (b,c) or an integer and twist is an integer</li>
<li><tt class="docutils literal"><span class="pre">object</span></tt> &#8211; a quiver mutation type</li>
</ul>
</blockquote>
<p>EXAMPLES:</p>
<p>Finite types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 5]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;B&#39;, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;B&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="go">[&#39;B&#39;, 5]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;B&#39;, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;C&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="go">[&#39;C&#39;, 5]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[ [&#39;A&#39;, 1], [&#39;A&#39;, 1] ]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 4]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 6]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 7]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">8</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;F&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="go">[&#39;F&#39;, 4]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;G&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;G&#39;, 2]</span>
</pre></div>
</div>
<p>Affine types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, [1, 1], 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;A&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, [2, 4], 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BB&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, [1, 1], 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BB&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BB&#39;, 2, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BB&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BB&#39;, 4, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CC&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, [1, 1], 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CC&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;CC&#39;, 2, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CC&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;CC&#39;, 4, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BC&#39;</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BC&#39;, 1, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BC&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BC&#39;, 5, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BD&#39;</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BD&#39;, 3, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BD&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;BD&#39;, 5, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CD&#39;</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;CD&#39;, 3, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CD&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;CD&#39;, 5, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 4, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;D&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 6, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 6, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">7</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 7, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8, 1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;F&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;F&#39;, 4, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;F&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;F&#39;, 4, -1]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;G&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;G&#39;, 2, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;G&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">[&#39;G&#39;, 2, -1]</span>
</pre></div>
</div>
<p>Elliptic types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[&#39;E&#39;, 6, [1, 1]]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">7</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[&#39;E&#39;, 7, [1, 1]]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">8</span><span class="p">,[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">])</span>
<span class="go">[&#39;E&#39;, 8, [1, 1]]</span>
</pre></div>
</div>
<p>Mutation finite types:</p>
<p>rank 2 cases:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;B&#39;, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;G&#39;, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;BC&#39;, 1, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;R2&#39;, [1, 5], 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, [1, 1], 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;R2&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;R2&#39;, [3, 5], 2]</span>
</pre></div>
</div>
<p>exceptional quiver mutation types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;V&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;V&#39;, 4, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;W&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;W&#39;, 4, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;W&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;W&#39;, 4, -2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;X&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;X&#39;, 6, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;Y&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;Y&#39;, 6, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;Z&#39;, 6, 2]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;Z&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">[&#39;Z&#39;, 6, -2]</span>
</pre></div>
</div>
<p>Mutation infinite types:</p>
<p>infinite type E:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">9</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">10</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 10, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;E&#39;</span><span class="p">,</span><span class="mi">12</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 12, 3]</span>

<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;AE&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;AE&#39;, [1, 1], 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;AE&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;AE&#39;, [1, 4], 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;BE&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;BE&#39;, 5, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;CE&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;CE&#39;, 5, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;DE&#39;</span><span class="p">,</span><span class="mi">6</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;DE&#39;, 6, 3]</span>
</pre></div>
</div>
<p>Grassmannian types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 4]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 6]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">8</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;GR&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;GR&#39;, [3, 10], 3]</span>
</pre></div>
</div>
<p>Triangular types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;TR&#39;</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;TR&#39;</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 6]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;TR&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8, [1, 1]]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;TR&#39;</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;TR&#39;, 5, 3]</span>
</pre></div>
</div>
<p><span class="math">T</span> types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 4]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;A&#39;, 7]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 4]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;D&#39;, 6]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 6]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 7]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">6</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 8, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 10, 3]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;E&#39;, 6, 1]</span>
<span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">(</span><span class="s">&#39;T&#39;</span><span class="p">,(</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">3</span><span class="p">)</span>
<span class="go">[&#39;T&#39;, [3, 3, 4], 3]</span>
</pre></div>
</div>
<p>Reducible types:</p>
<div class="highlight-python"><div class="highlight"><pre class="literal-block"><span class="gp">sage: </span><span class="n">QuiverMutationType</span><span class="p">([</span><span class="s">&#39;A&#39;</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="s">&#39;B&#39;</span><span class="p">,</span><span class="mi">4</span><span class="p">])</span>
<span class="go">[ [&#39;A&#39;, 3], [&#39;B&#39;, 4] ]</span>
</pre></div>
</div>


</div>
</html>
}}}

{{{id=135|
Gr = ClusterSeed(['GR',[4,9],3]); Gr; Gr.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|12|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['GR',|\phantom{x}\verb|[4,|\phantom{x}\verb|9],|\phantom{x}\verb|3]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=136|
Gr.is_mutation_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}</span></html>
}}}

{{{id=137|
Gr2 = ClusterSeed(['GR',[4,8],3]); Gr2;
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|9|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['E',|\phantom{x}\verb|7,|\phantom{x}\verb|[1,|\phantom{x}\verb|1]]|</span></html>
}}}

{{{id=139|
Gr2.is_mutation_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=140|
Tr = ClusterSeed(['TR',5,3]); Tr; Tr.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|15|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['TR',|\phantom{x}\verb|5,|\phantom{x}\verb|3]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=138|
NSL = ClusterSeed(['F',4,1]); NSL; NSL.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|5|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['F',|\phantom{x}\verb|4,|\phantom{x}\verb|1]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=141|
NSL2 = ClusterSeed(['F',4,-1]); NSL2; NSL2.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|5|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['F',|\phantom{x}\verb|4,|\phantom{x}\verb|-1]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=142|
BB = NSL.b_matrix_class(); len(BB);
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}60</span></html>
}}}

{{{id=144|
BB
///
WARNING: Output truncated!  
<html><a target='_new' href='/home/stumpc5/8/cells/144/full_output.txt' class='file_link'>full_output.txt</a></html>



<html><span class="math">\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]</span></html>
}}}

{{{id=143|
BB2 = NSL2.b_matrix_class(); len(BB2)
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}60</span></html>
}}}

{{{id=145|
BB2 = NSL2.b_matrix_class(up_to_equivalence=False); len(BB2)
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}720</span></html>
}}}

{{{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()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=99|
VC = EE7.variable_class(); len(VC)
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}70</span></html>
}}}

{{{id=54|
VC[35]
///
<html><span class="math">\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}}</span></html>
}}}

{{{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]
///
<html><span class="math">\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}}</span></html>
}}}

{{{id=67|
X = ClusterSeed(['X',6,2]); X; X.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|6|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['X',|\phantom{x}\verb|6,|\phantom{x}\verb|2]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=68|
S = ClusterSeed(['X',7,2]); S; S.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|7|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['X',|\phantom{x}\verb|7,|\phantom{x}\verb|2]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=71|
X.is_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}</span></html>
}}}

{{{id=69|
X.is_mutation_finite()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}</span></html>
}}}

{{{id=70|
V = ClusterSeed(['V',4,2]); V; V.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|4|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['V',|\phantom{x}\verb|4,|\phantom{x}\verb|2]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=72|
VMC = V.b_matrix_class(); len(VMC); VMC
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}7</span></html>
<html><span class="math">\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]</span></html>
}}}

{{{id=81|
WW = ClusterSeed(['W',4,2]); WW; WW.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|4|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['W',|\phantom{x}\verb|4,|\phantom{x}\verb|2]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=147|
WW2 = ClusterSeed(['W',4,-2]); WW2; WW2.show()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|4|\phantom{x}\verb|of|\phantom{x}\verb|type|\phantom{x}\verb|['W',|\phantom{x}\verb|4,|\phantom{x}\verb|-2]|</span></html>
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=148|
WW.b_matrix_class()
///
<html><span class="math">\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]</span></html>
}}}

{{{id=149|
WW2.b_matrix_class()
///
<html><span class="math">\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]</span></html>
}}}

{{{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
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\verb|A|\phantom{x}\verb|seed|\phantom{x}\verb|for|\phantom{x}\verb|a|\phantom{x}\verb|cluster|\phantom{x}\verb|algebra|\phantom{x}\verb|of|\phantom{x}\verb|rank|\phantom{x}\verb|4|</span></html>
}}}

{{{id=95|
Somos4.set_cluster([1,1,1,1])
///
}}}

{{{id=77|
Somos4.show()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
}}}

{{{id=151|
Somos4.mutate([0,1,2,3]); Somos4.show(); Somos4.cluster()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[2, 3, 7, 23\right]</span></html>
}}}

{{{id=152|
Somos4.mutate([0,1,2,3]); Somos4.show(); Somos4.cluster()
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[59, 314, 1529, 8209\right]</span></html>
}}}

{{{id=153|
Somos4.mutation_sequence([0,1,2,3,0,1,2,3],return_output='var')
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left[83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786\right]</span></html>
}}}

{{{id=154|

///
}}}

{{{id=159|

///
}}}

{{{id=160|

///
}}}

{{{id=162|

///
}}}