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<p><a id="X7C1FAB6280A02CCB" name="X7C1FAB6280A02CCB"></a></p>
<div class="ChapSects"><a href="chap74.html#X7C1FAB6280A02CCB">74 <span class="Heading">Unknowns</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap74.html#X85A1A27686C8D366">74.1 <span class="Heading">More about Unknowns</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap74.html#X79BAB8C48394779C">74.1-1 Unknown</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap74.html#X7B38F63581D7A96A">74.1-2 LargestUnknown</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap74.html#X828556067E069B6D">74.1-3 IsUnknown</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap74.html#X7E6B0D62788BB464">74.1-4 <span class="Heading">Comparison of Unknowns</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap74.html#X81EFCA7C82E18EFF">74.1-5 <span class="Heading">Arithmetical Operations for Unknowns</span></a>
</span>
</div></div>
</div>
<h3>74 <span class="Heading">Unknowns</span></h3>
<p>Sometimes the result of an operation does not allow further computations with it. In many cases, then an error is signalled, and the computation is stopped.</p>
<p>This is not appropriate for some applications in character theory. For example, if one wants to induce a character of a group to a supergroup (see <code class="func">InducedClassFunction</code> (<a href="chap72.html#X7FE39D3D78855D3B"><span class="RefLink">72.9-3</span></a>)) but the class fusion is only a parametrized map (see Chapter <a href="chap73.html#X7DF1ACDE7E9C6294"><span class="RefLink">73</span></a>), there may be values of the induced character which are determined by the fusion map, whereas other values are not known.</p>
<p>For this and other situations, <strong class="pkg">GAP</strong> provides the data type <em>unknown</em>. An object of this type, further on called an <em>unknown</em>, may stand for any cyclotomic (see Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a>), in particular its family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>) is <code class="code">CyclotomicsFamily</code>.</p>
<p>Unknowns are parametrized by positive integers. When a <strong class="pkg">GAP</strong> session is started, no unknowns exist.</p>
<p>The only ways to create unknowns are to call the function <code class="func">Unknown</code> (<a href="chap74.html#X79BAB8C48394779C"><span class="RefLink">74.1-1</span></a>) or a function that calls it, or to do arithmetical operations with unknowns.</p>
<p><strong class="pkg">GAP</strong> objects containing unknowns will contain <em>fixed</em> unknowns when they are printed to files, i.e., function calls <code class="code">Unknown(</code><span class="SimpleMath">n</span><code class="code">)</code> instead of <code class="code">Unknown()</code>. So be careful to read files printed in different <strong class="pkg">GAP</strong> sessions, since there may be the same unknown at different places.</p>
<p>The rest of this chapter contains information about the unknown constructor, the category, and comparison of and arithmetical operations for unknowns. More is not known about unknowns in <strong class="pkg">GAP</strong>.</p>
<p><a id="X85A1A27686C8D366" name="X85A1A27686C8D366"></a></p>
<h4>74.1 <span class="Heading">More about Unknowns</span></h4>
<p><a id="X79BAB8C48394779C" name="X79BAB8C48394779C"></a></p>
<h5>74.1-1 Unknown</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Unknown</code>( [<var class="Arg">n</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Called without argument, <code class="func">Unknown</code> returns a new unknown value, i.e., the first one that is larger than all unknowns which exist in the current <strong class="pkg">GAP</strong> session.</p>
<p>Called with a positive integer <var class="Arg">n</var>, <code class="func">Unknown</code> returns the <var class="Arg">n</var>-th unknown; if this did not exist yet, it is created.</p>
<p><a id="X7B38F63581D7A96A" name="X7B38F63581D7A96A"></a></p>
<h5>74.1-2 LargestUnknown</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestUnknown</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="func">LargestUnknown</code> is the largest <var class="Arg">n</var> that is used in any <code class="code">Unknown( <var class="Arg">n</var> )</code> in the current <strong class="pkg">GAP</strong> session. This is used in <code class="func">Unknown</code> (<a href="chap74.html#X79BAB8C48394779C"><span class="RefLink">74.1-1</span></a>) which increments this value when asked to make a new unknown.</p>
<p><a id="X828556067E069B6D" name="X828556067E069B6D"></a></p>
<h5>74.1-3 IsUnknown</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUnknown</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the category of unknowns in <strong class="pkg">GAP</strong>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown(); List( [ 1 .. 20 ], i -> Unknown() );;</span>
Unknown(1)
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown(); # note that we have already created 21 unknowns.</span>
Unknown(22)
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown(2000); Unknown();</span>
Unknown(2000)
Unknown(2001)
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestUnknown;</span>
2001
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUnknown( Unknown ); IsUnknown( Unknown() );</span>
false
true
</pre></div>
<p><a id="X7E6B0D62788BB464" name="X7E6B0D62788BB464"></a></p>
<h5>74.1-4 <span class="Heading">Comparison of Unknowns</span></h5>
<p>Unknowns can be <em>compared</em> via <code class="code">=</code> and <code class="code"><</code> with all cyclotomics and with certain other <strong class="pkg">GAP</strong> objects (see <a href="chap4.html#X7A274A1F8553B7E6"><span class="RefLink">4.12</span></a>). We have <code class="code">Unknown( <var class="Arg">n</var> ) >= Unknown( <var class="Arg">m</var> )</code> if and only if <code class="code"><var class="Arg">n</var> >= <var class="Arg">m</var></code> holds, unknowns are larger than all cyclotomics that are not unknowns.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown() >= Unknown(); Unknown(2) < Unknown(3);</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown() > 3; Unknown() > E(3);</span>
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Unknown() > Z(8); Unknown() > [];</span>
false
false
</pre></div>
<p><a id="X81EFCA7C82E18EFF" name="X81EFCA7C82E18EFF"></a></p>
<h5>74.1-5 <span class="Heading">Arithmetical Operations for Unknowns</span></h5>
<p>The usual arithmetic operations <code class="code">+</code>, <code class="code">-</code>, <code class="code">*</code> and <code class="code">/</code> are defined for addition, subtraction, multiplication and division of unknowns and cyclotomics. The result will be a new unknown except in one of the following cases.</p>
<p>Multiplication with zero yields zero, and multiplication with one or addition of zero yields the old unknown. <em>Note</em> that division by an unknown causes an error, since an unknown might stand for zero.</p>
<p>As unknowns are cyclotomics, dense lists of unknowns and other cyclotomics are row vectors and they can be added and multiplied in the usual way. Consequently, lists of such row vectors of equal length are (ordinary) matrices (see <code class="func">IsOrdinaryMatrix</code> (<a href="chap24.html#X7CF42B8A845BC6A9"><span class="RefLink">24.2-2</span></a>)).</p>
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