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<div class="ChapSects"><a href="chap5.html#X794C1A137F8FA14D">5 <span class="Heading">Methods for testing</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7DE7E7187BE24368">5.1 <span class="Heading">Testing</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X801680187A8CA462">5.1-1 SL2WithConjClasses</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A97C7C77C7608D0">5.1-2 SL2ChiST</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BFA24298556FF78">5.1-3 SL2TestPositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B41A0178465F138">5.1-4 SL2TestSymmetry</a></span>
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<h3>5 <span class="Heading">Methods for testing</span></h3>

<p>By the Chinese Remainder Theorem, it suffices to test irreps of prime power level, so those are the irreps handled by the functions in this section.</p>

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<h4>5.1 <span class="Heading">Testing</span></h4>

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<h5>5.1-1 SL2WithConjClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2WithConjClasses</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the group <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span> with conjugacy classes set to the format we use.</p>

<p><a id="X7A97C7C77C7608D0" name="X7A97C7C77C7608D0"></a></p>

<h5>5.1-2 SL2ChiST</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2ChiST</code>( <var class="Arg">S</var>, <var class="Arg">T</var>, <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list representing a character of <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>.</p>

<p>Converts the modular data <span class="Math">(S,T)</span>, which must have level dividing <span class="Math">p^\lambda</span>, into a character of <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>, presented in a form matching the conjugacy classes used in <code class="code">SL2WithConjClasses</code>.</p>

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<h5>5.1-3 SL2TestPositions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2TestPositions</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a boolean.</p>

<p>Constructs and tests all non-trivial irreps of level dividing <span class="Math">p^\lambda</span> by checking their positions in <code class="code">Irr(G)</code> (see <span class="URL"><a href="https://www.gap-system.org/Manuals/doc/ref/chap71.html#X873B3CC57E9A5492">Section 71.8-2 of the GAP Manual</a></span>). Note that this function will print information on the irreps involved if <code class="code">InfoSL2Reps</code> is set to level 1 or higher; see Section <a href="chap1.html#X86A9B6F87E619FFF"><span class="RefLink">1.2</span></a>.</p>

<p><a id="X7B41A0178465F138" name="X7B41A0178465F138"></a></p>

<h5>5.1-4 SL2TestSymmetry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2TestSymmetry</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a boolean.</p>

<p>Constructs and tests all irreps of level <span class="Math">p^\lambda</span>, confirming that the <span class="Math">S</span>-matrix is symmetric and unitary and the <span class="Math">T</span> matrix is diagonal. Note that this function will print information on the irreps involved if <code class="code">InfoSL2Reps</code> is set to level 1 or higher; see Section <a href="chap1.html#X86A9B6F87E619FFF"><span class="RefLink">1.2</span></a>.</p>


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