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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap72.html">[Previous Chapter]</a> <a href="chap74.html">[Next Chapter]</a> </div>
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<p><a id="X7DF1ACDE7E9C6294" name="X7DF1ACDE7E9C6294"></a></p>
<div class="ChapSects"><a href="chap73.html#X7DF1ACDE7E9C6294">73 <span class="Heading">Maps Concerning Character Tables</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X7FED949A86575949">73.1 <span class="Heading">Power Maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X781FAA497E3B4D1A">73.1-1 PowerMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7C7B292E80590BE0">73.1-2 PossiblePowerMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7E0289957E9D62EE">73.1-3 ElementOrdersPowerMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7C0F171F7DC846B7">73.1-4 PowerMapByComposition</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X80980FF37F0D521B">73.2 <span class="Heading">Orbits on Sets of Possible Power Maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7ECB9DDE8608B9A9">73.2-1 OrbitPowerMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X8753F5217A570529">73.2-2 RepresentativesPowerMaps</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X806975FE81534444">73.3 <span class="Heading">Class Fusions between Character Tables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X86CE53B681F13C63">73.3-1 <span class="Heading">FusionConjugacyClasses</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7F71402285B7DE8E">73.3-2 ComputedClassFusions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X8464DD23879431D9">73.3-3 GetFusionMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X808970FE87C3432F">73.3-4 StoreFusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7F6569D5786A9D49">73.3-5 NamesOfFusionSources</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7883271F7F26356E">73.3-6 PossibleClassFusions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7BCC5B4B7E9DF42C">73.3-7 ConsiderStructureConstants</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X7C34060278E4BFC4">73.4 <span class="Heading">Orbits on Sets of Possible Class Fusions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X79A0FE1C853302D2">73.4-1 OrbitFusions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X821D11D180B5D317">73.4-2 RepresentativesFusions</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X7F18772E86F06179">73.5 <span class="Heading">Parametrized Maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X8740C1397C6A96C8">73.5-1 CompositionMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7877EE167A711AB6">73.5-2 InverseMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X82C0E76F804C3FF7">73.5-3 ProjectionMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7D9CA09385467EDE">73.5-4 Indirected</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7910BE5687DDAAF3">73.5-5 Parametrized</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7917265684700B10">73.5-6 ContainedMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X80C7328C85BFC20B">73.5-7 UpdateMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X81A1A0E88570E42A">73.5-8 MeetMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X8593A72A8193EC8B">73.5-9 CommutativeDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7B6EC10C7F7411E9">73.5-10 CheckFixedPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7AD5158E82AF1CD4">73.5-11 TransferDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X78487F03852A503B">73.5-12 TestConsistencyMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7DAD6EA585D74615">73.5-13 Indeterminateness</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7888BDC88304BE5A">73.5-14 PrintAmbiguity</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7F957B1481E10A0C">73.5-15 ContainedSpecialVectors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X84F87C2282EFB0EE">73.5-16 CollapsedMat</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X81F1137A874EB962">73.5-17 ContainedDecomposables</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X86472A217D6C3CE7">73.6 <span class="Heading">Subroutines for the Construction of Power Maps</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X85D068D77C3C041C">73.6-1 InitPowerMap</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7B27749E7BF54EBB">73.6-2 Congruences</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7D31B1548205E222">73.6-3 ConsiderKernels</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7DD1DCF3865E0017">73.6-4 ConsiderSmallerPowerMaps</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X805B6C1C78AA5DB6">73.6-5 MinusCharacter</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X808CCF6087D5B661">73.6-6 PowerMapsAllowedBySymmetrizations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap73.html#X7AF7305D80E1E5EF">73.7 <span class="Heading">Subroutines for the Construction of Class Fusions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7E2BC50C86A16604">73.7-1 InitFusion</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X82F776A3850C6404">73.7-2 CheckPermChar</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X7C52CEDB7D98A6B8">73.7-3 ConsiderTableAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap73.html#X85024BAE8585DB1C">73.7-4 FusionsAllowedByRestrictions</a></span>
</div></div>
</div>
<h3>73 <span class="Heading">Maps Concerning Character Tables</span></h3>
<p>Besides the characters, <em>power maps</em> are an important part of a character table, see Section <a href="chap73.html#X7FED949A86575949"><span class="RefLink">73.1</span></a>. Often their computation is not easy, and if the table has no access to the underlying group then in general they cannot be obtained from the matrix of irreducible characters; so it is useful to store them on the table.</p>
<p>If not only a single table is considered but different tables of a group and a subgroup or of a group and a factor group are used, also <em>class fusion maps</em> (see Section <a href="chap73.html#X806975FE81534444"><span class="RefLink">73.3</span></a>) must be known to get information about the embedding or simply to induce or restrict characters, see Section <a href="chap72.html#X854A4E3A85C5F89B"><span class="RefLink">72.9</span></a>).</p>
<p>These are examples of functions from conjugacy classes which will be called <em>maps</em> in the following. (This should not be confused with the term mapping, cf. Chapter <a href="chap32.html#X7C9734B880042C73"><span class="RefLink">32</span></a>.) In <strong class="pkg">GAP</strong>, maps are represented by lists. Also each character, each list of element orders, of centralizer orders, or of class lengths are maps, and the list returned by <code class="func">ListPerm</code> (<a href="chap42.html#X7A9DCFD986958C1E"><span class="RefLink">42.5-1</span></a>), when this function is called with a permutation of classes, is a map.</p>
<p>When maps are constructed without access to a group, often one only knows that the image of a given class is contained in a set of possible images, e. g., that the image of a class under a subgroup fusion is in the set of all classes with the same element order. Using further information, such as centralizer orders, power maps and the restriction of characters, the sets of possible images can be restricted further. In many cases, at the end the images are uniquely determined.</p>
<p>Because of this approach, many functions in this chapter work not only with maps but with <em>parametrized maps</em> (or <em>paramaps</em> for short). More about parametrized maps can be found in Section <a href="chap73.html#X7F18772E86F06179"><span class="RefLink">73.5</span></a>.</p>
<p>The implementation follows <a href="chapBib.html#biBBre91">[Bre91]</a>, a description of the main ideas together with several examples can be found in <a href="chapBib.html#biBBre99">[Bre99]</a>.</p>
<p>Several examples in this chapter require the <strong class="pkg">GAP</strong> Character Table Library to be available. If it is not yet loaded then we load it now.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage( "ctbllib" );</span>
true
</pre></div>
<p><a id="X7FED949A86575949" name="X7FED949A86575949"></a></p>
<h4>73.1 <span class="Heading">Power Maps</span></h4>
<p>The <span class="SimpleMath">n</span>-th power map of a character table is represented by a list that stores at position <span class="SimpleMath">i</span> the position of the class containing the <span class="SimpleMath">n</span>-th powers of the elements in the <span class="SimpleMath">i</span>-th class. The <span class="SimpleMath">n</span>-th power map can be composed from the power maps of the prime divisors of <span class="SimpleMath">n</span>, so usually only power maps for primes are actually stored in the character table.</p>
<p>For an ordinary character table <var class="Arg">tbl</var> with access to its underlying group <span class="SimpleMath">G</span>, the <span class="SimpleMath">p</span>-th power map of <var class="Arg">tbl</var> can be computed using the identification of the conjugacy classes of <span class="SimpleMath">G</span> with the classes of <var class="Arg">tbl</var>. For an ordinary character table without access to a group, in general the <span class="SimpleMath">p</span>-th power maps (and hence also the element orders) for prime divisors <span class="SimpleMath">p</span> of the group order are not uniquely determined by the matrix of irreducible characters. So only necessary conditions can be checked in this case, which in general yields only a list of several possibilities for the desired power map. Character tables of the <strong class="pkg">GAP</strong> character table library store all <span class="SimpleMath">p</span>-th power maps for prime divisors <span class="SimpleMath">p</span> of the group order.</p>
<p>Power maps of Brauer tables can be derived from the power maps of the underlying ordinary tables.</p>
<p>For (computing and) accessing the <span class="SimpleMath">n</span>-th power map of a character table, <code class="func">PowerMap</code> (<a href="chap73.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) can be used; if the <span class="SimpleMath">n</span>-th power map cannot be uniquely determined then <code class="func">PowerMap</code> (<a href="chap73.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) returns <code class="keyw">fail</code>.</p>
<p>The list of all possible <span class="SimpleMath">p</span>-th power maps of a table in the sense that certain necessary conditions are satisfied can be computed with <code class="func">PossiblePowerMaps</code> (<a href="chap73.html#X7C7B292E80590BE0"><span class="RefLink">73.1-2</span></a>). This provides a default strategy, the subroutines are listed in Section <a href="chap73.html#X86472A217D6C3CE7"><span class="RefLink">73.6</span></a>.</p>
<p><a id="X781FAA497E3B4D1A" name="X781FAA497E3B4D1A"></a></p>
<h5>73.1-1 PowerMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PowerMap</code>( <var class="Arg">tbl</var>, <var class="Arg">n</var>[, <var class="Arg">class</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PowerMapOp</code>( <var class="Arg">tbl</var>, <var class="Arg">n</var>[, <var class="Arg">class</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComputedPowerMaps</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Called with first argument a character table <var class="Arg">tbl</var> and second argument an integer <var class="Arg">n</var>, <code class="func">PowerMap</code> returns the <var class="Arg">n</var>-th power map of <var class="Arg">tbl</var>. This is a list containing at position <span class="SimpleMath">i</span> the position of the class of <var class="Arg">n</var>-th powers of the elements in the <span class="SimpleMath">i</span>-th class of <var class="Arg">tbl</var>.</p>
<p>If the additional third argument <var class="Arg">class</var> is present then the position of <var class="Arg">n</var>-th powers of the <var class="Arg">class</var>-th class is returned.</p>
<p>If the <var class="Arg">n</var>-th power map is not uniquely determined by <var class="Arg">tbl</var> then <code class="keyw">fail</code> is returned. This can happen only if <var class="Arg">tbl</var> has no access to its underlying group.</p>
<p>The power maps of <var class="Arg">tbl</var> that were computed already by <code class="func">PowerMap</code> are stored in <var class="Arg">tbl</var> as value of the attribute <code class="func">ComputedPowerMaps</code>, the <span class="SimpleMath">n</span>-th power map at position <span class="SimpleMath">n</span>. <code class="func">PowerMap</code> checks whether the desired power map is already stored, computes it using the operation <code class="func">PowerMapOp</code> if it is not yet known, and stores it. So methods for the computation of power maps can be installed for the operation <code class="func">PowerMapOp</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "L3(2)" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ComputedPowerMaps( tbl );</span>
[ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],,,,
[ 1, 2, 3, 4, 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMap( tbl, 5 );</span>
[ 1, 2, 3, 4, 6, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ComputedPowerMaps( tbl );</span>
[ , [ 1, 1, 3, 2, 5, 6 ], [ 1, 2, 1, 4, 6, 5 ],, [ 1, 2, 3, 4, 6, 5 ],
, [ 1, 2, 3, 4, 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMap( tbl, 137, 2 );</span>
2
</pre></div>
<p><a id="X7C7B292E80590BE0" name="X7C7B292E80590BE0"></a></p>
<h5>73.1-2 PossiblePowerMaps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PossiblePowerMaps</code>( <var class="Arg">tbl</var>, <var class="Arg">p</var>[, <var class="Arg">options</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>For the ordinary character table <var class="Arg">tbl</var> of a group <span class="SimpleMath">G</span> and a prime integer <var class="Arg">p</var>, <code class="func">PossiblePowerMaps</code> returns the list of all maps that have the following properties of the <span class="SimpleMath">p</span>-th power map of <var class="Arg">tbl</var>. (Representative orders are used only if the <code class="func">OrdersClassRepresentatives</code> (<a href="chap71.html#X86F455DA7A9C30EE"><span class="RefLink">71.9-1</span></a>) value of <var class="Arg">tbl</var> is known.</p>
<ol>
<li><p>For class <span class="SimpleMath">i</span>, the centralizer order of the image is a multiple of the <span class="SimpleMath">i</span>-th centralizer order; if the elements in the <span class="SimpleMath">i</span>-th class have order coprime to <span class="SimpleMath">p</span> then the centralizer orders of class <span class="SimpleMath">i</span> and its image are equal.</p>
</li>
<li><p>Let <span class="SimpleMath">n</span> be the order of elements in class <span class="SimpleMath">i</span>. If <var class="Arg">prime</var> divides <span class="SimpleMath">n</span> then the images have order <span class="SimpleMath">n/p</span>; otherwise the images have order <span class="SimpleMath">n</span>. These criteria are checked in <code class="func">InitPowerMap</code> (<a href="chap73.html#X85D068D77C3C041C"><span class="RefLink">73.6-1</span></a>).</p>
</li>
<li><p>For each character <span class="SimpleMath">χ</span> of <span class="SimpleMath">G</span> and each element <span class="SimpleMath">g</span> in <span class="SimpleMath">G</span>, the values <span class="SimpleMath">χ(g^p)</span> and <code class="code">GaloisCyc</code><span class="SimpleMath">( χ(g), p )</span> are algebraic integers that are congruent modulo <span class="SimpleMath">p</span>; if <span class="SimpleMath">p</span> does not divide the element order of <span class="SimpleMath">g</span> then the two values are equal. This congruence is checked for the characters specified below in the discussion of the <var class="Arg">options</var> argument; For linear characters <span class="SimpleMath">λ</span> among these characters, the condition <span class="SimpleMath">χ(g)^p = χ(g^p)</span> is checked. The corresponding function is <code class="func">Congruences</code> (<a href="chap73.html#X7B27749E7BF54EBB"><span class="RefLink">73.6-2</span></a>).</p>
</li>
<li><p>For each character <span class="SimpleMath">χ</span> of <span class="SimpleMath">G</span>, the kernel is a normal subgroup <span class="SimpleMath">N</span>, and <span class="SimpleMath">g^p ∈ N</span> for all <span class="SimpleMath">g ∈ N</span>; moreover, if <span class="SimpleMath">N</span> has index <span class="SimpleMath">p</span> in <span class="SimpleMath">G</span> then <span class="SimpleMath">g^p ∈ N</span> for all <span class="SimpleMath">g ∈ G</span>, and if the index of <span class="SimpleMath">N</span> in <span class="SimpleMath">G</span> is coprime to <span class="SimpleMath">p</span> then <span class="SimpleMath">g^p not ∈ N</span> for each <span class="SimpleMath">g not ∈ N</span>. These conditions are checked for the kernels of all characters <span class="SimpleMath">χ</span> specified below, the corresponding function is <code class="func">ConsiderKernels</code> (<a href="chap73.html#X7D31B1548205E222"><span class="RefLink">73.6-3</span></a>).</p>
</li>
<li><p>If <span class="SimpleMath">p</span> is larger than the order <span class="SimpleMath">m</span> of an element <span class="SimpleMath">g ∈ G</span> then the class of <span class="SimpleMath">g^p</span> is determined by the power maps for primes dividing the residue of <span class="SimpleMath">p</span> modulo <span class="SimpleMath">m</span>. If these power maps are stored in the <code class="func">ComputedPowerMaps</code> (<a href="chap73.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) value of <var class="Arg">tbl</var> then this information is used. This criterion is checked in <code class="func">ConsiderSmallerPowerMaps</code> (<a href="chap73.html#X7DD1DCF3865E0017"><span class="RefLink">73.6-4</span></a>).</p>
</li>
<li><p>For each character <span class="SimpleMath">χ</span> of <span class="SimpleMath">G</span>, the symmetrization <span class="SimpleMath">ψ</span> defined by <span class="SimpleMath">ψ(g) = (χ(g)^p - χ(g^p))/p</span> is a character. This condition is checked for the kernels of all characters <span class="SimpleMath">χ</span> specified below, the corresponding function is <code class="func">PowerMapsAllowedBySymmetrizations</code> (<a href="chap73.html#X808CCF6087D5B661"><span class="RefLink">73.6-6</span></a>).</p>
</li>
</ol>
<p>If <var class="Arg">tbl</var> is a Brauer table, the possibilities are computed from those for the underlying ordinary table.</p>
<p>The optional argument <var class="Arg">options</var>, if given, must be a record that may have the following components:</p>
<dl>
<dt><strong class="Mark"><code class="code">chars</code>:</strong></dt>
<dd><p>a list of characters which are used for the check of the criteria 3., 4., and 6.; the default is <code class="code">Irr( <var class="Arg">tbl</var> )</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">powermap</code>:</strong></dt>
<dd><p>a parametrized map which is an approximation of the desired map</p>
</dd>
<dt><strong class="Mark"><code class="code">decompose</code>:</strong></dt>
<dd><p>a Boolean; a <code class="keyw">true</code> value indicates that all constituents of the symmetrizations of <code class="code">chars</code> computed for criterion 6. lie in <code class="code">chars</code>, so the symmetrizations can be decomposed into elements of <code class="code">chars</code>; the default value of <code class="code">decompose</code> is <code class="keyw">true</code> if <code class="code">chars</code> is not bound and <code class="code">Irr( <var class="Arg">tbl</var> )</code> is known, otherwise <code class="keyw">false</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">quick</code>:</strong></dt>
<dd><p>a Boolean; if <code class="keyw">true</code> then the subroutines are called with value <code class="keyw">true</code> for the argument <var class="Arg">quick</var>; especially, as soon as only one candidate remains this candidate is returned immediately; the default value is <code class="keyw">false</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">parameters</code>:</strong></dt>
<dd><p>a record with components <code class="code">maxamb</code>, <code class="code">minamb</code> and <code class="code">maxlen</code> which control the subroutine <code class="func">PowerMapsAllowedBySymmetrizations</code> (<a href="chap73.html#X808CCF6087D5B661"><span class="RefLink">73.6-6</span></a>); it only uses characters with current indeterminateness up to <code class="code">maxamb</code>, tests decomposability only for characters with current indeterminateness at least <code class="code">minamb</code>, and admits a branch according to a character only if there is one with at most <code class="code">maxlen</code> possible symmetrizations.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "U4(3).4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PossiblePowerMaps( tbl, 2 );</span>
[ [ 1, 1, 3, 4, 5, 2, 2, 8, 3, 4, 11, 12, 6, 14, 9, 1, 1, 2, 2, 3, 4,
5, 6, 8, 9, 9, 10, 11, 12, 16, 16, 16, 16, 17, 17, 18, 18, 18,
18, 20, 20, 20, 20, 22, 22, 24, 24, 25, 26, 28, 28, 29, 29 ] ]
</pre></div>
<p><a id="X7E0289957E9D62EE" name="X7E0289957E9D62EE"></a></p>
<h5>73.1-3 ElementOrdersPowerMap</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementOrdersPowerMap</code>( <var class="Arg">powermap</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">powermap</var> be a nonempty list containing at position <span class="SimpleMath">p</span>, if bound, the <span class="SimpleMath">p</span>-th power map of a character table or group. <code class="func">ElementOrdersPowerMap</code> returns a list of the same length as each entry in <var class="Arg">powermap</var>, with entry at position <span class="SimpleMath">i</span> equal to the order of elements in class <span class="SimpleMath">i</span> if this order is uniquely determined by <var class="Arg">powermap</var>, and equal to an unknown (see Chapter <a href="chap74.html#X7C1FAB6280A02CCB"><span class="RefLink">74</span></a>) otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "U4(3).4" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">known:= ComputedPowerMaps( tbl );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( known );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">sub:= ShallowCopy( known );; Unbind( sub[7] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementOrdersPowerMap( sub );</span>
[ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, Unknown(1), Unknown(2), 8, 9, 12, 2,
2, 4, 4, 6, 6, 6, 8, 10, 12, 12, 12, Unknown(3), Unknown(4), 4, 4,
4, 4, 4, 4, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 20, 20, 24, 24,
Unknown(5), Unknown(6), Unknown(7), Unknown(8) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ord:= ElementOrdersPowerMap( known );</span>
[ 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 12, 2, 2, 4, 4, 6, 6, 6,
8, 10, 12, 12, 12, 14, 14, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 12, 12,
12, 12, 12, 12, 20, 20, 24, 24, 28, 28, 28, 28 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ord = OrdersClassRepresentatives( tbl );</span>
true
</pre></div>
<p><a id="X7C0F171F7DC846B7" name="X7C0F171F7DC846B7"></a></p>
<h5>73.1-4 PowerMapByComposition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PowerMapByComposition</code>( <var class="Arg">tbl</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">tbl</var> must be a nearly character table, and <var class="Arg">n</var> a positive integer. If the power maps for all prime divisors of <var class="Arg">n</var> are stored in the <code class="func">ComputedPowerMaps</code> (<a href="chap73.html#X781FAA497E3B4D1A"><span class="RefLink">73.1-1</span></a>) list of <var class="Arg">tbl</var> then <code class="func">PowerMapByComposition</code> returns the <var class="Arg">n</var>-th power map of <var class="Arg">tbl</var>. Otherwise <code class="keyw">fail</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "U4(3).4" );; exp:= Exponent( tbl );</span>
2520
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMapByComposition( tbl, exp );</span>
[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( ComputedPowerMaps( tbl ) );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMapByComposition( tbl, 11 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMap( tbl, 11 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PowerMapByComposition( tbl, 11 );</span>
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 26, 25, 27, 28, 29, 31, 30, 33, 32, 35, 34, 37,
36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52 ]
</pre></div>
<p><a id="X80980FF37F0D521B" name="X80980FF37F0D521B"></a></p>
<h4>73.2 <span class="Heading">Orbits on Sets of Possible Power Maps</span></h4>
<p>The permutation group of matrix automorphisms (see <code class="func">MatrixAutomorphisms</code> (<a href="chap71.html#X84353BB884AF0365"><span class="RefLink">71.22-1</span></a>)) acts on the possible power maps returned by <code class="func">PossiblePowerMaps</code> (<a href="chap73.html#X7C7B292E80590BE0"><span class="RefLink">73.1-2</span></a>) by permuting a list via <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-17</span></a>) and then mapping the images via <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>). Note that by definition, the group of <em>table</em> automorphisms acts trivially.</p>
<p><a id="X7ECB9DDE8608B9A9" name="X7ECB9DDE8608B9A9"></a></p>
<h5>73.2-1 OrbitPowerMaps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrbitPowerMaps</code>( <var class="Arg">map</var>, <var class="Arg">permgrp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the orbit of the power map <var class="Arg">map</var> under the action of the permutation group <var class="Arg">permgrp</var> via a combination of <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-17</span></a>) and <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>).</p>
<p><a id="X8753F5217A570529" name="X8753F5217A570529"></a></p>
<h5>73.2-2 RepresentativesPowerMaps</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativesPowerMaps</code>( <var class="Arg">listofmaps</var>, <var class="Arg">permgrp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of orbit representatives of the power maps in the list <var class="Arg">listofmaps</var> under the action of the permutation group <var class="Arg">permgrp</var> via a combination of <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-17</span></a>) and <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">tbl:= CharacterTable( "3.McL" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">grp:= MatrixAutomorphisms( Irr( tbl ) ); Size( grp );</span>
<permutation group with 5 generators>
32
<span class="GAPprompt">gap></span> <span class="GAPinput">poss:= PossiblePowerMaps( CharacterTable( "3.McL" ), 3 );</span>
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:= RepresentativesPowerMaps( poss, grp );</span>
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">orb:= OrbitPowerMaps( reps[1], grp );</span>
[ [ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 8, 9, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ],
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, 9, 8, 37,
37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52, 52, 49, 49,
49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Parametrized( orb );</span>
[ 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 11, 11, 11, 14, 14, 14, 17, 17, 17,
4, 4, 4, 4, 4, 4, 29, 29, 29, 26, 26, 26, 32, 32, 32, [ 8, 9 ],
[ 8, 9 ], 37, 37, 37, 40, 40, 40, 43, 43, 43, 11, 11, 11, 52, 52,
52, 49, 49, 49, 14, 14, 14, 14, 14, 14, 37, 37, 37, 37, 37, 37 ]
</pre></div>
<p><a id="X806975FE81534444" name="X806975FE81534444"></a></p>
<h4>73.3 <span class="Heading">Class Fusions between Character Tables</span></h4>
<p>For a group <span class="SimpleMath">G</span> and a subgroup <span class="SimpleMath">H</span> of <span class="SimpleMath">G</span>, the fusion map between the character table of <span class="SimpleMath">H</span> and the character table of <span class="SimpleMath">G</span> is represented by a list that stores at position <span class="SimpleMath">i</span> the position of the <span class="SimpleMath">i</span>-th class of the table of <span class="SimpleMath">H</span> in the classes list of the table of <span class="SimpleMath">G</span>.</p>
<p>For ordinary character tables <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var> of <span class="SimpleMath">H</span> and <span class="SimpleMath">G</span>, with access to the groups <span class="SimpleMath">H</span> and <span class="SimpleMath">G</span>, the class fusion between <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var> can be computed using the identifications of the conjugacy classes of <span class="SimpleMath">H</span> with the classes of <var class="Arg">tbl1</var> and the conjugacy classes of <span class="SimpleMath">G</span> with the classes of <var class="Arg">tbl2</var>. For two ordinary character tables without access to an underlying group, or in the situation that the group stored in <var class="Arg">tbl1</var> is not physically a subgroup of the group stored in <var class="Arg">tbl2</var> but an isomorphic copy, in general the class fusion is not uniquely determined by the information stored on the tables such as irreducible characters and power maps. So only necessary conditions can be checked in this case, which in general yields only a list of several possibilities for the desired class fusion. Character tables of the <strong class="pkg">GAP</strong> character table library store various class fusions that are regarded as important, for example fusions from maximal subgroups (see <code class="func">ComputedClassFusions</code> (<a href="chap73.html#X7F71402285B7DE8E"><span class="RefLink">73.3-2</span></a>) and <code class="func">Maxes</code> (<a href="../../pkg/ctbllib/doc/chap3.html#X8150E63F7DBDF252"><span class="RefLink">CTblLib: Maxes</span></a>) in the manual for the <strong class="pkg">GAP</strong> Character Table Library).</p>
<p>Class fusions between Brauer tables can be derived from the class fusions between the underlying ordinary tables. The class fusion from a Brauer table to the underlying ordinary table is stored when the Brauer table is constructed from the ordinary table, so no method is needed to compute such a fusion.</p>
<p>For (computing and) accessing the class fusion between two character tables, <code class="func">FusionConjugacyClasses</code> (<a href="chap73.html#X86CE53B681F13C63"><span class="RefLink">73.3-1</span></a>) can be used; if the class fusion cannot be uniquely determined then <code class="func">FusionConjugacyClasses</code> (<a href="chap73.html#X86CE53B681F13C63"><span class="RefLink">73.3-1</span></a>) returns <code class="keyw">fail</code>.</p>
<p>The list of all possible class fusion between two tables in the sense that certain necessary conditions are satisfied can be computed with <code class="func">PossibleClassFusions</code> (<a href="chap73.html#X7883271F7F26356E"><span class="RefLink">73.3-6</span></a>). This provides a default strategy, the subroutines are listed in Section <a href="chap73.html#X7AF7305D80E1E5EF"><span class="RefLink">73.7</span></a>.</p>
<p>It should be noted that all the following functions except <code class="func">FusionConjugacyClasses</code> (<a href="chap73.html#X86CE53B681F13C63"><span class="RefLink">73.3-1</span></a>) deal only with the situation of class fusions from subgroups. The computation of <em>factor fusions</em> from a character table to the table of a factor group is not dealt with here. Since the ordinary character table of a group <span class="SimpleMath">G</span> determines the character tables of all factor groups of <span class="SimpleMath">G</span>, the factor fusion to a given character table of a factor group of <span class="SimpleMath">G</span> is determined up to table automorphisms (see <code class="func">AutomorphismsOfTable</code> (<a href="chap71.html#X7C2753DE8094F4BA"><span class="RefLink">71.9-4</span></a>)) once the class positions of the kernel of the natural epimorphism have been fixed.</p>
<p><a id="X86CE53B681F13C63" name="X86CE53B681F13C63"></a></p>
<h5>73.3-1 <span class="Heading">FusionConjugacyClasses</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionConjugacyClasses</code>( <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionConjugacyClasses</code>( <var class="Arg">H</var>, <var class="Arg">G</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionConjugacyClasses</code>( <var class="Arg">hom</var>[, <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionConjugacyClassesOp</code>( <var class="Arg">tbl1</var>, <var class="Arg">tbl2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FusionConjugacyClassesOp</code>( <var class="Arg">hom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Called with two character tables <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>, <code class="func">FusionConjugacyClasses</code> returns the fusion of conjugacy classes between <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>. (If one of the tables is a Brauer table, it will delegate this task to the underlying ordinary table.)</p>
<p>Called with two groups <var class="Arg">H</var> and <var class="Arg">G</var> where <var class="Arg">H</var> is a subgroup of <var class="Arg">G</var>, <code class="func">FusionConjugacyClasses</code> returns the fusion of conjugacy classes between <var class="Arg">H</var> and <var class="Arg">G</var>. This is done by delegating to the ordinary character tables of <var class="Arg">H</var> and <var class="Arg">G</var>, since class fusions are stored only for character tables and not for groups.</p>
<p>Note that the returned class fusion refers to the ordering of conjugacy classes in the character tables if the arguments are character tables and to the ordering of conjugacy classes in the groups if the arguments are groups (see <code class="func">ConjugacyClasses</code> (<a href="chap71.html#X849A38F887F6EC86"><span class="RefLink">71.6-2</span></a>)).</p>
<p>Called with a group homomorphism <var class="Arg">hom</var>, <code class="func">FusionConjugacyClasses</code> returns the fusion of conjugacy classes between the preimage and the image of <var class="Arg">hom</var>; contrary to the two cases above, also factor fusions can be handled by this variant. If <var class="Arg">hom</var> is the only argument then the class fusion refers to the ordering of conjugacy classes in the groups. If the character tables of preimage and image are given as <var class="Arg">tbl1</var> and <var class="Arg">tbl2</var>, respectively (each table with its group stored), then the fusion refers to the ordering of classes in these tables.</p>
<p>If no class fusion exists or if the class fusion is not uniquely determined, <code class="keyw">fail</code> is returned; this may happen when <code class="func">FusionConjugacyClasses</code> is called with two character tables that do not know compatible underlying groups.</p>
<p>Methods for the computation of class fusions can be installed for the operation <code class="func">FusionConjugacyClassesOp</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s4:= SymmetricGroup( 4 );</span>
Sym( [ 1 .. 4 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">tbls4:= CharacterTable( s4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">d8:= SylowSubgroup( s4, 2 );</span>
Group([ (1,2), (3,4), (1,3)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">FusionConjugacyClasses( d8, s4 );</span>
[ 1, 2, 3, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">tbls5:= CharacterTable( "S5" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">FusionConjugacyClasses( CharacterTable( "A5" ), tbls5 );</span>
[ 1, 2, 3, 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FusionConjugacyClasses(CharacterTable("A5"), CharacterTable("J1"));</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">PossibleClassFusions(CharacterTable("A5"), CharacterTable("J1"));</span>
[ [ 1, 2, 3, 4, 5 ], [ 1, 2, 3, 5, 4 ] ]
</pre></div>
<p><a id="X7F71402285B7DE8E" name="X7F71402285B7DE8E"></a></p>
<h5>73.3-2 ComputedClassFusions</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComputedClassFusions</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The class fusions from the character table <var class="Arg">tbl</var> that have been computed already by <code class="func">FusionConjugacyClasses</code> (<a href="chap73.html#X86CE53B681F13C63"><span class="RefLink">73.3-1</span></a>) or explicitly stored by <code class="func">StoreFusion</code> (<a href="chap73.html#X808970FE87C3432F"><span class="RefLink">73.3-4</span></a>) are stored in the <code class="func">ComputedClassFusions</code> list of <var class="Arg">tbl1</var>. Each entry of this list is a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the <code class="func">Identifier</code> (<a href="chap71.html#X79C40EE97890202F"><span class="RefLink">71.9-8</span></a>) value of the character table to which the fusion maps,</p>
</dd>
<dt><strong class="Mark"><code class="code">map</code></strong></dt>
<dd><p>the list of positions of image classes,</p>
</dd>
<dt><strong class="Mark"><code class="code">text</code> (optional)</strong></dt>
<dd><p>a string giving additional information about the fusion map, for example whether the map is uniquely determined by the character tables,</p>
</dd>
<dt><strong class="Mark"><code class="code">specification</code> (optional, rarely used)</strong></dt>
<dd><p>a value that distinguishes different fusions between the same tables.</p>
</dd>
</dl>
<p>Note that stored fusion maps may differ from the maps returned by <code class="func">GetFusionMap</code> (<a href="chap73.html#X8464DD23879431D9"><span class="RefLink">73.3-3</span></a>) and the maps entered by <code class="func">StoreFusion</code> (<a href="chap73.html#X808970FE87C3432F"><span class="RefLink">73.3-4</span></a>) if the table <var class="Arg">destination</var> has a nonidentity <code class="func">ClassPermutation</code> (<a href="chap71.html#X8099FEDC7DE03AEE"><span class="RefLink">71.21-5</span></a>) value. So if one fetches a fusion map from a table <var class="Arg">tbl1</var> to a table <var class="Arg">tbl2</var> via access to the data in the <code class="func">ComputedClassFusions</code> list of <var class="Arg">tbl1</var> then the stored value must be composed with the <code class="func">ClassPermutation</code> (<a href="chap71.html#X8099FEDC7DE03AEE"><span class="RefLink">71.21-5</span></a>) value of <var class="Arg">tbl2</var> in order to obtain the correct class fusion. (If one handles fusions only via <code class="func">GetFusionMap</code> (<a href="chap73.html#X8464DD23879431D9"><span class="RefLink">73.3-3</span></a>) and <code class="func">StoreFusion</code> (<a href="chap73.html#X808970FE87C3432F"><span class="RefLink">73.3-4</span></a>) then this adjustment is made automatically.)</p>
<p>Fusions are identified via the <code class="func">Identifier</code> (<a href="chap71.html#X79C40EE97890202F"><span class="RefLink">71.9-8</span></a>) value of the destination table and not by this table itself because many fusions between character tables in the <strong class="pkg">GAP</strong> character table library are stored on library tables, and it is not desirable to load together with a library | |
