https://github.com/uberparagon/mgn
Tip revision: bebc795e6778db315f0b7144c07024f8a666722f authored by Drew Johnson on 12 September 2017, 18:57:37 UTC
maybe some changes
maybe some changes
Tip revision: bebc795
index.html
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>strataalgebra Documentation — StrataAlgegra 1.0.0 documentation</title>
<link rel="stylesheet" href="_static/nature.css" type="text/css" />
<link rel="stylesheet" href="_static/pygments.css" type="text/css" />
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: './',
VERSION: '1.0.0',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true,
SOURCELINK_SUFFIX: '.txt'
};
</script>
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
</head>
<body role="document">
<div class="related" role="navigation" aria-label="related navigation">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
accesskey="I">index</a></li>
<li class="right" >
<a href="py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="nav-item nav-item-0"><a href="#">StrataAlgegra 1.0.0 documentation</a> »</li>
</ul>
</div>
<div class="document">
<div class="documentwrapper">
<div class="bodywrapper">
<div class="body" role="main">
<div class="toctree-wrapper compound">
</div>
<div class="section" id="strataalgebra-documentation">
<h1>strataalgebra Documentation<a class="headerlink" href="#strataalgebra-documentation" title="Permalink to this headline">¶</a></h1>
<p>Welcome! The <code class="docutils literal"><span class="pre">strataalgebra</span></code> module is designed to compute products in the strata algebra using sage. The strata algebra is of interest
because it has a natural surjective morphism to the tautological ring of the moduli space of curves. The kernel is described by the Faber-Zagier relations.</p>
<p>The product structure was implemented by <a class="reference external" href="http://pages.uoregon.edu/drewj/">Drew Johnson</a>, based on <a class="reference external" href="https://arxiv.org/abs/0808.1974">S. Yang’s note</a> describing the algorithm of Graber and
Pandharipande.
The program also computes the FZ relations using code copied from A. Pixton’s <a class="reference external" href="http://math.mit.edu/~apixton/programs/tautrel.sage">tautrel.sage</a> program.
The code was integrated into this package (with some modifications and optimizations) by Drew Johnson
(who takes responsibility for any bugs introduced!).</p>
</div>
<div class="section" id="installation">
<h1>Installation<a class="headerlink" href="#installation" title="Permalink to this headline">¶</a></h1>
<p>Installation should be (hopefully) easy. <code class="docutils literal"><span class="pre">strataalgebra</span></code> is distributed as part of the <a class="reference external" href="https://github.com/uberparagon/mgn">mgn package</a>.</p>
<p>Once it is installed, you can load it in a sage session by typing:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
</pre></div>
</div>
</div>
<div class="section" id="how-to-use">
<h1>How to use<a class="headerlink" href="#how-to-use" title="Permalink to this headline">¶</a></h1>
<dl class="class">
<dt id="strataalgebra.StrataAlgebra">
<em class="property">class </em><code class="descclassname">strataalgebra.</code><code class="descname">StrataAlgebra</code><span class="sig-paren">(</span><em>base</em>, <em>g</em>, <em>markings=()</em>, <em>make_vars=True</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra" title="Permalink to this definition">¶</a></dt>
<dd><p>A ring representing the Strata algebra.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
<li><strong>base</strong> (<em>Ring</em>) – The ring of coefficients you want to work over, usually <cite>QQ</cite>.</li>
<li><strong>g</strong> (<em>int</em>) – The genus.</li>
<li><strong>markings</strong> (<em>tuple</em>) – The markings. They should be positive integers. Repeats are allowed. Defaults to no markings.</li>
<li><strong>make_vars</strong> (<em>bool</em>) – Defaults to True. If True, creates variables ps, ps2, ka1, ... , ka{d} (where d is the dimension of the moduli space) that can be used to create basis elements.</li>
</ul>
</td>
</tr>
</tbody>
</table>
<p>First import the module:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
</pre></div>
</div>
<p>Construct a <code class="xref py py-class docutils literal"><span class="pre">StrataAlgebra</span></code>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="n">SA</span>
<span class="go">Strata algebra with genus 1 and markings (1, 2) over Rational Field</span>
</pre></div>
</div>
<p>Print the basis in a certain codimension</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">print_strata</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">**** i: 0</span>
<span class="go">[0 1 2 0 0]</span>
<span class="go">[0 0 0 2 1]</span>
<span class="go">[0 1 1 0 1]</span>
<span class="go">**** i: 1</span>
<span class="go">[0 1 2 0 0]</span>
<span class="go">[0 0 1 1 1]</span>
<span class="go">[0 1 0 1 1]</span>
<span class="go">**** i: 2</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ka1 1 1 2]</span>
<span class="go">**** i: 3</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 1 1 ps + 2]</span>
<span class="go">**** i: 4</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 ps + 1 1 2]</span>
<span class="go">**** i: 5</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 1 ps + 1 2]</span>
<span class="go">**** i: 6</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 1 1 1]</span>
<span class="go">[ka1 + 1 0 0 1]</span>
<span class="go">**** i: 7</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 1 1 1]</span>
<span class="go">[ 1 0 0 ps + 1]</span>
<span class="go">**** i: 8</span>
<span class="go">ka2</span>
<span class="go">**** i: 9</span>
<span class="go">ka1^2</span>
<span class="go">**** i: 10</span>
<span class="go">ka1*ps1</span>
<span class="go">**** i: 11</span>
<span class="go">ka1*ps2</span>
<span class="go">**** i: 12</span>
<span class="go">ps1^2</span>
<span class="go">**** i: 13</span>
<span class="go">ps1*ps2</span>
<span class="go">**** i: 14</span>
<span class="go">ps2^2</span>
</pre></div>
</div>
<p>Classes that are monomials in <span class="math">\(\psi\)</span> and <span class="math">\(\kappa\)</span> have self-explanatory names.</p>
<p>More complicated classes are displayed in a matrix as follows:</p>
<p>Each row after the first corresponds to a vertex of the graph.
The constant term in the entry in the first column is the genus.
The kappa classes also appear in the first column.
Each column beyond the first corresponds to an edge or a half edge.
The entry in the first row gives the label of the half edge, or 0 for a full edge.
The constant term of the entry in location (v,e) gives the number of times (0, 1, or 2) that edge e touches vertex v.
A <cite>ps</cite> in entry (v,e) means a <span class="math">\(\psi\)</span>-class associated to the half edge coming out of v.
A <cite>ps_</cite> may occur when there is a loop at the vertex.
The entry in the top left is just padding.</p>
<p>To create classes, you can use their codimension and index.
Boundary strata and psi-kappa monomials are represented by their special names,
and the rest are represented by <cite>s_{codim},{index}</cite></p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">a</span> <span class="o">=</span> <span class="n">SA</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="n">a</span>
<span class="go">s_2,1</span>
<span class="gp">sage: </span><span class="n">b</span> <span class="o">=</span> <span class="n">SA</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">7</span><span class="p">);</span> <span class="n">b</span>
<span class="go">s_2,7</span>
<span class="gp">sage: </span><span class="n">c</span> <span class="o">=</span> <span class="n">SA</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">11</span><span class="p">);</span> <span class="n">c</span>
<span class="go">ka1*ps2</span>
<span class="gp">sage: </span><span class="n">d</span> <span class="o">=</span> <span class="n">SA</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="n">d</span>
<span class="go">Dg1</span>
</pre></div>
</div>
<p>Vector space arithmetic is supported.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="mi">3</span><span class="o">*</span><span class="n">b</span>
<span class="go">3*s_2,7</span>
<span class="gp">sage: </span><span class="n">a</span><span class="o">*</span><span class="mi">72</span> <span class="o">-</span> <span class="n">b</span><span class="o">/</span><span class="mi">17</span>
<span class="go">72*s_2,1 + -1/17*s_2,7</span>
</pre></div>
</div>
<p>Use <a class="reference internal" href="#strataalgebra.StrataAlgebra.get_stratum" title="strataalgebra.StrataAlgebra.get_stratum"><code class="xref py py-meth docutils literal"><span class="pre">get_stratum()</span></code></a> if you need to know what an unamed basis element means.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">get_stratum</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span>
<span class="go">[ 0 1 2 0]</span>
<span class="go">[ 0 1 1 1]</span>
<span class="go">[ 1 0 0 ps + 1]</span>
</pre></div>
</div>
<p>You can construct <span class="math">\(\psi,\;\kappa\)</span> monomials and boundary divisors with the methods <a class="reference internal" href="#strataalgebra.StrataAlgebra.kappa" title="strataalgebra.StrataAlgebra.kappa"><code class="xref py py-meth docutils literal"><span class="pre">kappa()</span></code></a>, <a class="reference internal" href="#strataalgebra.StrataAlgebra.psi" title="strataalgebra.StrataAlgebra.psi"><code class="xref py py-meth docutils literal"><span class="pre">psi()</span></code></a>,
<a class="reference internal" href="#strataalgebra.StrataAlgebra.boundary" title="strataalgebra.StrataAlgebra.boundary"><code class="xref py py-meth docutils literal"><span class="pre">boundary()</span></code></a>, and <a class="reference internal" href="#strataalgebra.StrataAlgebra.irr" title="strataalgebra.StrataAlgebra.irr"><code class="xref py py-meth docutils literal"><span class="pre">irr()</span></code></a>.</p>
<p>You can construct an element using the matrix notation. Just pass a list of lists into your
<code class="xref py py-class docutils literal"><span class="pre">StrataAlgebra</span></code>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">var</span><span class="p">(</span><span class="s1">'ps'</span><span class="p">)</span> <span class="c1">#This is usually be done automatically, but we have to do it manually for the dotests to work.</span>
<span class="go">ps</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="p">([[</span><span class="mi">0</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">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">ps</span><span class="o">+</span><span class="mi">1</span><span class="p">],[</span><span class="mi">0</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">s_2,7</span>
</pre></div>
</div>
<p>Here is an example of the <cite>ps_</cite>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span><span class="mi">2</span><span class="p">,())</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">get_stratum</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">9</span><span class="p">)</span>
<span class="go">[ 0 0]</span>
<span class="go">[ 1 ps^2 + 2]</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">get_stratum</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="go">[ 0 0]</span>
<span class="go">[ 1 ps*ps_ + 2]</span>
<span class="gp">sage: </span><span class="n">var</span><span class="p">(</span><span class="s1">'ps_'</span><span class="p">)</span>
<span class="go">ps_</span>
<span class="gp">sage: </span><span class="n">s</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],[</span><span class="mi">1</span><span class="p">,</span><span class="n">ps_</span><span class="o">*</span><span class="n">ps</span><span class="o">+</span><span class="mi">2</span><span class="p">]])</span>
<span class="go">s_3,10</span>
</pre></div>
</div>
<p>One of the main features is the computation of the product.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">a</span><span class="o">*</span><span class="n">b</span>
<span class="go">0</span>
</pre></div>
</div>
<p>Of course, the codimension was too big. Lets do some less trivial ones.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="mi">3</span><span class="p">));</span> <span class="n">SA</span>
<span class="go">Strata algebra with genus 1 and markings (1, 2, 3) over Rational Field</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">SA</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">ps1*ps2</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">get_stratum</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span> <span class="c1">#just so you can see what it is</span>
<span class="go">[0 1 2 3 0 0]</span>
<span class="go">[0 0 1 1 1 0]</span>
<span class="go">[0 1 0 0 1 1]</span>
<span class="go">[1 0 0 0 0 1]</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">0</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">7</span><span class="p">)</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">s_3,36</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">get_stratum</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">36</span><span class="p">)</span>
<span class="go">[ 0 1 2 3 0 0]</span>
<span class="go">[ 0 0 1 1 1 0]</span>
<span class="go">[ 0 1 0 0 1 1]</span>
<span class="go">[ka1 + 1 0 0 0 0 1]</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">irr</span><span class="p">()</span><span class="o">^</span><span class="mi">3</span>
<span class="go">6*s_3,4 + -3*s_3,15 + -3*s_3,23 + -3*s_3,35 + s_3,48 + s_3,49</span>
</pre></div>
</div>
<p>Everything should work with distributive laws, etc.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">3</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">-</span><span class="n">SA</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="o">==</span> <span class="o">-</span><span class="n">SA</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="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">SA</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mi">3</span>
<span class="go">True</span>
</pre></div>
</div>
<p>It should work over any ring.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">R</span><span class="o">.<</span><span class="n">t</span><span class="o">></span> <span class="o">=</span> <span class="n">PolynomialRing</span><span class="p">(</span><span class="n">ZZ</span><span class="p">)</span>
<span class="gp">sage: </span><span class="n">SA</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="mi">2</span><span class="p">);</span> <span class="n">SA</span>
<span class="go">Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring</span>
<span class="gp">sage: </span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="n">t</span><span class="p">)</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">t</span><span class="o">^</span><span class="mi">2</span> <span class="o">+</span> <span class="n">t</span><span class="o">*</span><span class="n">SA</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">(t^2)one + (2*t + 3)ka1</span>
<span class="gp">sage: </span><span class="n">_</span><span class="o">^</span><span class="mi">2</span>
<span class="go">(t^4)one + (4*t^3 + 6*t^2)ka1 + (4*t^2 + 12*t + 9)ka1^2</span>
</pre></div>
</div>
<p>There may be problems over a non-divisible ring.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">irr</span><span class="p">()</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">TypeError</span>: <span class="n">unsupported operand parent(s) for *: 'Rational Field' and 'Strata algebra with genus 2 and markings () over Univariate Polynomial Ring in t over Integer Ring'</span>
</pre></div>
</div>
<p>Also, repeated names of the markings are allowed. The following corresponds to <span class="math">\(\overline{\mathcal M}_{1,2}/S_2\)</span>.
Compare the codimension 2 strata to our earlier example.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">SA</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="n">SA</span>
<span class="go">Strata algebra with genus 1 and markings (1, 1) over Rational Field</span>
<span class="gp">sage: </span><span class="n">SA</span><span class="o">.</span><span class="n">print_strata</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">**** i: 0</span>
<span class="go">[0 1 1 0 0]</span>
<span class="go">[0 0 0 2 1]</span>
<span class="go">[0 1 1 0 1]</span>
<span class="go">**** i: 1</span>
<span class="go">[0 1 1 0 0]</span>
<span class="go">[0 0 1 1 1]</span>
<span class="go">[0 1 0 1 1]</span>
<span class="go">**** i: 2</span>
<span class="go">[ 0 1 1 0]</span>
<span class="go">[ka1 1 1 2]</span>
<span class="go">**** i: 3</span>
<span class="go">[ 0 1 1 0]</span>
<span class="go">[ 0 1 1 ps + 2]</span>
<span class="go">**** i: 4</span>
<span class="go">[ 0 1 1 0]</span>
<span class="go">[ 0 ps + 1 1 2]</span>
<span class="go">**** i: 5</span>
<span class="go">[ 0 1 1 0]</span>
<span class="go">[ 0 1 1 1]</span>
<span class="go">[ka1 + 1 0 0 1]</span>
<span class="go">**** i: 6</span>
<span class="go">[ 0 1 1 0]</span>
<span class="go">[ 0 1 1 1]</span>
<span class="go">[ 1 0 0 ps + 1]</span>
<span class="go">**** i: 7</span>
<span class="go">ka2</span>
<span class="go">**** i: 8</span>
<span class="go">ka1^2</span>
<span class="go">**** i: 9</span>
<span class="go">ka1*ps1</span>
<span class="go">**** i: 10</span>
<span class="go">ps1^2</span>
<span class="go">**** i: 11</span>
<span class="go">ps1*ps1</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p><code class="xref py py-meth docutils literal"><span class="pre">strataalgebra.StrataAlgebraElement.integrate()</span></code></p>
<p><a class="reference internal" href="#strataalgebra.StrataAlgebra.betti" title="strataalgebra.StrataAlgebra.betti"><code class="xref py py-meth docutils literal"><span class="pre">betti()</span></code></a></p>
<p class="last"><a class="reference internal" href="#strataalgebra.StrataAlgebra.FZ_betti" title="strataalgebra.StrataAlgebra.FZ_betti"><code class="xref py py-meth docutils literal"><span class="pre">FZ_betti()</span></code></a></p>
</div>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.FZ_betti">
<code class="descname">FZ_betti</code><span class="sig-paren">(</span><em>codim=None</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.FZ_betti" title="Permalink to this definition">¶</a></dt>
<dd><p>Give the dimension of the cohomology of moduli space of curves, as predicted by the Faber-Zagier-Pixton relations.
If the codimension is omited, return a list of all of them.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>codim</strong> – Optional. The codimension you want.</td>
</tr>
</tbody>
</table>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,))</span><span class="o">.</span><span class="n">FZ_betti</span><span class="p">()</span>
<span class="go">[1, 3, 5, 3, 1]</span>
<span class="gp">sage: </span><span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,))</span><span class="o">.</span><span class="n">FZ_betti</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">5</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="#strataalgebra.StrataAlgebra.betti" title="strataalgebra.StrataAlgebra.betti"><code class="xref py py-meth docutils literal"><span class="pre">betti()</span></code></a></p>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.FZ_matrix">
<code class="descname">FZ_matrix</code><span class="sig-paren">(</span><em>r</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.FZ_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the matrix of Faber-Zagier-Pixton relations.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>r</strong> – </td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-class docutils literal"><span class="pre">Matrix</span></code></td>
</tr>
</tbody>
</table>
<p>The columns correspond to the basis elements of the Strata algebra, and each row is a relation.</p>
<p>Notice that this matrix considers the kappa classes to be in the monomial basis. Thus, is different than the
output of Pixton’s original <cite>taurel.sage</cite> program.</p>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="#strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis" title="strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis"><code class="xref py py-meth docutils literal"><span class="pre">FZ_matrix_pushforward_basis()</span></code></a></p>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis">
<code class="descname">FZ_matrix_pushforward_basis</code><span class="sig-paren">(</span><em>r</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.FZ_matrix_pushforward_basis" title="Permalink to this definition">¶</a></dt>
<dd><p>Return the matrix of Faber-Zagier-Pixton relations, using the “pushforward” basis, NOT the kappa monomial basis
that the rest of the code uses.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>r</strong> – </td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-class docutils literal"><span class="pre">Matrix</span></code></td>
</tr>
</tbody>
</table>
<p>The columns correspond to the basis elements of the Strata algebra, and each row is a relation.
This matrix should be the same as Pixton’s original <cite>tautrel.sage</cite> program after permuting columns.</p>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.MgnLb_class">
<code class="descname">MgnLb_class</code><span class="sig-paren">(</span><em>index</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.MgnLb_class" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the class corresponding to the index from Carl Faber’s Mgn_Lb Maple program.
This is useful for testing purposes.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_class</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">ps1</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_class</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="go">ka2</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_class</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
<span class="go">1/2*D_irr</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_class</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">ps2</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.MgnLb_int">
<code class="descname">MgnLb_int</code><span class="sig-paren">(</span><em>index_list</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.MgnLb_int" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes an integral of boundary divisors, kappa classes, and psi classes.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>index_list</strong> – A list of indices of classes, according the the scheme of Carl Faber’s MgnLb Maple program. This function is useful because so you can test our implementation of the product and the FZ_relations.</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><cite>Rational</cite></td>
</tr>
</tbody>
</table>
<p>Examples:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_int</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">6</span><span class="p">])</span>
<span class="go">1/2</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_int</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">1/24</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">MgnLb_int</span><span class="p">([</span><span class="mi">4</span><span class="p">])</span>
<span class="go">1/24</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="first admonition-title">See also</p>
<p class="last"><a class="reference internal" href="#strataalgebra.StrataAlgebra.MgnLb_class" title="strataalgebra.StrataAlgebra.MgnLb_class"><code class="xref py py-meth docutils literal"><span class="pre">MgnLb_class()</span></code></a></p>
</div>
</dd></dl>
<dl class="attribute">
<dt id="strataalgebra.StrataAlgebra.basis_integrals">
<code class="descname">basis_integrals</code><a class="headerlink" href="#strataalgebra.StrataAlgebra.basis_integrals" title="Permalink to this definition">¶</a></dt>
<dd><p>File: /Users/drewjohnson/mgn/strataalgebra/strataalgebra.py (starting at line 417)</p>
<p>Return a list of numbers corresponding to the integrals of the basis elements in the top codimension.</p>
<p>This is computed via the FZ relations, so it is probably not fast. However, it is a nice check.</p>
<p>The value is cached, so you only have to compute it once per session.</p>
<p>This is used by <code class="xref py py-meth docutils literal"><span class="pre">StrataAlgebraElement.integrate()</span></code> which is the more likely way you will want to use it.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">print_strata</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">**** i: 0</span>
<span class="go">D_irr</span>
<span class="go">**** i: 1</span>
<span class="go">ka1</span>
<span class="go">**** i: 2</span>
<span class="go">ps1</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">basis_integrals</span><span class="p">()</span>
<span class="go">[1, 1/24, 1/24]</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.betti">
<code class="descname">betti</code><span class="sig-paren">(</span><em>codim=None</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.betti" title="Permalink to this definition">¶</a></dt>
<dd><p>Give the number of basis elements in the Strata algebra for the given codimension.
If the codimension is omitted, return a list of all of them.</p>
<p>This is NOT the Betti numbers for the moduli space of curves! For that see <a class="reference internal" href="#strataalgebra.StrataAlgebra.FZ_betti" title="strataalgebra.StrataAlgebra.FZ_betti"><code class="xref py py-meth docutils literal"><span class="pre">FZ_betti()</span></code></a>.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>codim</strong> (<em>int</em>) – Optional. The codimension you want.</td>
</tr>
</tbody>
</table>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,))</span><span class="o">.</span><span class="n">betti</span><span class="p">()</span>
<span class="go">[1, 4, 17, 49, 92]</span>
<span class="gp">sage: </span><span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,))</span><span class="o">.</span><span class="n">betti</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">17</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.boundary">
<code class="descname">boundary</code><span class="sig-paren">(</span><em>g1</em>, <em>markings1=()</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.boundary" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a boundary divisor with genus <cite>g1</cite> and <cite>markings1</cite> points on one component</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>g1</strong> (<em>int</em>) – The genus on one component</li>
<li><strong>markings1</strong> (<em>list</em>) – A list or tuple of markings on the one component. Defaults to the empty list.</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="xref py py-class docutils literal"><span class="pre">StrataAlgebraElement</span></code></p>
</td>
</tr>
</tbody>
</table>
<p>Note that if the two components are the same, it will return the stratum with a coefficient of 1/2.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</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">3</span><span class="p">))</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">boundary</span><span class="p">(</span><span class="mi">0</span><span class="p">,</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">Dg0m1_2</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">boundary</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">Dg0m1_2</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">boundary</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">1/2*Dg1</span>
</pre></div>
</div>
<p>Each component must still be stable.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">boundary</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">KeyError</span>: <span class="n">Dg0</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.do_all_products">
<code class="descname">do_all_products</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.do_all_products" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute all the products of basis elements of the ring (thus caching their values).
This could take a long time for some rings.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">do_all_products</span><span class="p">()</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.get_stratum">
<code class="descname">get_stratum</code><span class="sig-paren">(</span><em>r</em>, <em>j</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.get_stratum" title="Permalink to this definition">¶</a></dt>
<dd><p>Get the <code class="xref py py-class docutils literal"><span class="pre">StrataGraph</span></code> associated a a codimension and an index.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
<li><strong>r</strong> – The codimension</li>
<li><strong>j</strong> – The index</li>
</ul>
</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last"><code class="xref py py-class docutils literal"><span class="pre">StrataGraph</span></code></p>
</td>
</tr>
</tbody>
</table>
<p>See <code class="xref py py-class docutils literal"><span class="pre">StrataAlgebra</span></code> documentation for examples.</p>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.irr">
<code class="descname">irr</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.irr" title="Permalink to this definition">¶</a></dt>
<dd><p>Make the irreducible boundary. It will be returned with a coefficient of 1/2.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">irr</span><span class="p">()</span>
<span class="go">1/2*D_irr</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.kappa">
<code class="descname">kappa</code><span class="sig-paren">(</span><em>a</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.kappa" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a kappa class.</p>
<table class="docutils field-list" frame="void" rules="none">
<col class="field-name" />
<col class="field-body" />
<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>a</strong> (<em>int</em>) – The subscript (codimension) of the kappa class you want.</td>
</tr>
<tr class="field-even field"><th class="field-name">Return type:</th><td class="field-body"><code class="xref py py-class docutils literal"><span class="pre">StrataAlgebraElement</span></code></td>
</tr>
</tbody>
</table>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span><span class="mi">2</span><span class="p">,())</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">ka1</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="go">ka3</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">ka1*ka2</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">0</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">boundary</span><span class="p">(</span><span class="mi">1</span><span class="p">,())</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">s_2,4</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">get_stratum</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="go">[ 0 0]</span>
<span class="go">[ka1 + 1 1]</span>
<span class="go">[ 1 1]</span>
</pre></div>
</div>
<p>The subscript must be less than or equal to the dimension of the moduli space.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">kappa</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">KeyError</span>: <span class="n">4</span>
</pre></div>
</div>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.print_strata">
<code class="descname">print_strata</code><span class="sig-paren">(</span><em>r</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.print_strata" title="Permalink to this definition">¶</a></dt>
<dd><p>Print all the strata, with their indexes, in codimension <cite>r</cite>.
:param r: The codimension
:return: None</p>
<p>See <code class="xref py py-class docutils literal"><span class="pre">StrataAlgebra</span></code> documentation for examples.</p>
</dd></dl>
<dl class="method">
<dt id="strataalgebra.StrataAlgebra.psi">
<code class="descname">psi</code><span class="sig-paren">(</span><em>mark</em><span class="sig-paren">)</span><a class="headerlink" href="#strataalgebra.StrataAlgebra.psi" title="Permalink to this definition">¶</a></dt>
<dd><p>Return a psi class.
:param int mark: The mark that the <span class="math">\(\psi\)</span>-class is associated to.
:rtype: <code class="xref py py-class docutils literal"><span class="pre">StrataAlgebraElement</span></code></p>
<p>Examples</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">from</span> <span class="nn">strataalgebra</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span><span class="mi">0</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">3</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
<span class="gp">sage: </span><span class="n">psi1</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">psi1</span>
<span class="go">ps1</span>
<span class="gp">sage: </span><span class="n">psi2</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">psi2</span>
<span class="go">ps2</span>
<span class="gp">sage: </span><span class="n">psi1</span><span class="o">*</span><span class="n">psi2</span>
<span class="go">ps1*ps2</span>
<span class="gp">sage: </span><span class="p">(</span><span class="n">psi1</span><span class="o">*</span><span class="n">psi2</span><span class="p">)</span><span class="o">.</span><span class="n">integrate</span><span class="p">()</span>
<span class="go">2</span>
</pre></div>
</div>
<p>Note that the psi variables are not automatically injected into the namespace</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">ps1</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">NameError</span>: <span class="n">name 'ps1' is not defined</span>
</pre></div>
</div>
<p>In case of repeated marks, notice that the <span class="math">\(\psi\)</span>-class is the pushforward under the quotient map. Observe:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>
<span class="go">ps1^2 + ps1*ps1</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="gt">Traceback (most recent call last):</span>
<span class="c">...</span>
<span class="gr">ValueError</span>: <span class="n">tuple.index(x): x not in tuple</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</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">1</span><span class="p">))</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>
<span class="go">2*ps1^2 + 4*ps1*ps1</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">^</span><span class="mi">3</span>
<span class="go">4*ps1^3 + 24*ps1^2*ps1 + 8*ps1*ps1*ps1</span>
<span class="gp">sage: </span><span class="n">var</span><span class="p">(</span><span class="s1">'ps'</span><span class="p">)</span> <span class="c1">#This should be done automatically.</span>
<span class="go">ps</span>
<span class="gp">sage: </span><span class="n">s</span><span class="p">([[</span><span class="mi">0</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">1</span><span class="p">,</span><span class="n">ps</span><span class="o">^</span><span class="mi">3</span><span class="o">+</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">ps1^3</span>
<span class="gp">sage: </span><span class="n">s</span> <span class="o">=</span> <span class="n">StrataAlgebra</span><span class="p">(</span><span class="n">QQ</span><span class="p">,</span><span class="mi">0</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">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">12*ps1*ps2</span>
<span class="gp">sage: </span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">s</span><span class="o">.</span><span class="n">psi</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="go">4*ps1^2 + 8*ps1*ps1</span>
</pre></div>
</div>
<p>This maybe looks surprising, but it makes sense with the formula</p>
<div class="math">
\[\pi_*(\alpha) \pi_*(\beta) = \frac{1}{|G|} \pi_* \left( \sum_{\sigma, \tau \in G} \sigma_*(\alpha) \tau_*(\beta) \right)\]</div>
<p>for two class <span class="math">\(\alpha,\;\beta \in H^*(X)\)</span> where <span class="math">\(\pi: X \rightarrow X/G\)</span> is the quotient map and <span class="math">\(G\)</span> is a finite group.</p>
</dd></dl>
</dd></dl>
</div>
<div class="section" id="module-tests">
<span id="testing"></span><h1>Testing<a class="headerlink" href="#module-tests" title="Permalink to this headline">¶</a></h1>
<p>The <code class="docutils literal"><span class="pre">tests</span></code> module tests the implementation of the product and the FZ relations by comparing them to the <code class="docutils literal"><span class="pre">topintersections</span></code> code.</p>
<p>You can change which pairs you want to check by modifying the <code class="docutils literal"><span class="pre">g_n_pairs_to_check</span> <span class="pre">variable</span></code> in the source code <code class="docutils literal"><span class="pre">tests.py</span></code>.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">sage: </span><span class="kn">import</span> <span class="nn">strataalgebra.tests</span> <span class="kn">as</span> <span class="nn">tests</span>
<span class="gp">sage: </span><span class="n">tests</span><span class="o">.</span><span class="n">run</span><span class="p">()</span> <span class="c1"># long time</span>
</pre></div>
</div>
</div>
<div class="section" id="indices-and-tables">
<h1>Indices and tables<a class="headerlink" href="#indices-and-tables" title="Permalink to this headline">¶</a></h1>
<ul class="simple">
<li><a class="reference internal" href="genindex.html"><span class="std std-ref">Index</span></a></li>
<li><a class="reference internal" href="py-modindex.html"><span class="std std-ref">Module Index</span></a></li>
<li><a class="reference internal" href="search.html"><span class="std std-ref">Search Page</span></a></li>
</ul>
</div>
</div>
</div>
</div>
<div class="sphinxsidebar" role="navigation" aria-label="main navigation">
<div class="sphinxsidebarwrapper">
<h3><a href="#">Table Of Contents</a></h3>
<ul>
<li><a class="reference internal" href="#">strataalgebra Documentation</a></li>
<li><a class="reference internal" href="#installation">Installation</a></li>
<li><a class="reference internal" href="#how-to-use">How to use</a></li>
<li><a class="reference internal" href="#module-tests">Testing</a></li>
<li><a class="reference internal" href="#indices-and-tables">Indices and tables</a></li>
</ul>
<div role="note" aria-label="source link">
<h3>This Page</h3>
<ul class="this-page-menu">
<li><a href="_sources/index.rst.txt"
rel="nofollow">Show Source</a></li>
</ul>
</div>
<div id="searchbox" style="display: none" role="search">
<h3>Quick search</h3>
<form class="search" action="search.html" method="get">
<div><input type="text" name="q" /></div>
<div><input type="submit" value="Go" /></div>
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
<script type="text/javascript">$('#searchbox').show(0);</script>
</div>
</div>
<div class="clearer"></div>
</div>
<div class="related" role="navigation" aria-label="related navigation">
<h3>Navigation</h3>
<ul>
<li class="right" style="margin-right: 10px">
<a href="genindex.html" title="General Index"
>index</a></li>
<li class="right" >
<a href="py-modindex.html" title="Python Module Index"
>modules</a> |</li>
<li class="nav-item nav-item-0"><a href="#">StrataAlgegra 1.0.0 documentation</a> »</li>
</ul>
</div>
<div class="footer" role="contentinfo">
© Copyright 2017, Drew Johnson.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.5.3.
</div>
</body>
</html>