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<div class="ChapSects"><a href="chap6_mj.html#X803E01577A2B37D2">6 <span class="Heading">Automorphisms of Groupoids</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7935832881320667">6.1 <span class="Heading">Automorphisms of single piece groupoids</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X79DA704A8051CB72">6.1-1 GroupoidAutomorphismByObjectPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X86BD61567B41B139">6.1-2 GroupoidInnerAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8278B93E7F78038C">6.1-3 <span class="Heading">Automorphisms of a groupoid with rays</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7DB9D5737EB8C260">6.1-4 AutomorphismGroupOfGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7EC237ED7E1978B0">6.1-5 <span class="Heading">Inner automorphisms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X80F9594481CA9FDE">6.1-6 GroupoidAutomorphismByGroupAutos</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A93C6507BC697CD">6.1-7 AutomorphismGroupoidOfGroupoid</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X8003AF117B956D16">6.2 <span class="Heading">Matrix representations of groupoids</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7DB8E7EF7A51F1BE">6.3 <span class="Heading">Groupoid actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7C4DCB287B3DD0CD">6.3-1 GroupoidActionByConjugation</a></span>
</div></div>
</div>
<h3>6 <span class="Heading">Automorphisms of Groupoids</span></h3>
<p>In this chapter we consider automorphisms of single piece groupoids; then homogeneous discrete groupoids; and finally homogeneous groupoids. We also consider matrix representations and groupooid actions.</p>
<p><a id="X7935832881320667" name="X7935832881320667"></a></p>
<h4>6.1 <span class="Heading">Automorphisms of single piece groupoids</span></h4>
<p><a id="X79DA704A8051CB72" name="X79DA704A8051CB72"></a></p>
<h5>6.1-1 GroupoidAutomorphismByObjectPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupoidAutomorphismByObjectPerm</code>( <var class="Arg">gpd</var>, <var class="Arg">imobs</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">&#an style='color: green'>8227; GroupoidAutomorphismByGroupAuto</code>( <var class="Arg">gpd</var>, <var class="Arg">gpiso</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">&#an style='color: green'>8227; GroupoidAutomorphismByNtuple</code>( <var class="Arg">gpd</var>, <var class="Arg">imrays</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">&#an style='color: green'>8227; GroupoidAutomorphismByRayShifts</code>( <var class="Arg">gpd</var>, <var class="Arg">imrays</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>We first describe automorphisms of a groupoid <span class="SimpleMath">\(G\)</span> where <span class="SimpleMath">\(G\)</span> is the direct product of a group <span class="SimpleMath">\(g\)</span> and a complete digraph with <span class="SimpleMath">\(n\)</span> objects.. The automorphism group is generated by three types of automorphism:</p>
<ul>
<li><p>given a permutation <span class="SimpleMath">\(\pi\)</span> of the <span class="SimpleMath">\(n\)</span> objects, we define</p>
<p class="center">\[
\alpha_{\pi} : G \to G,~
(g : o_i \to o_j) \mapsto (g : o_{\pi i} \to o_{\pi j});
\]</p>
</li>
<li><p>given an automorphism <span class="SimpleMath">\(\theta\)</span> of the root group <span class="SimpleMath">\(g\)</span>, we define</p>
<p class="center">\[
\alpha_{\theta} : G \to G,~ (g : o_i \to o_j) \mapsto (\theta g : o_i \to o_j);
\]</p>
</li>
<li><p>given <span class="SimpleMath">\(L = [g_1,g_2,g_3,\ldots,g_n] \in g^n\)</span> we define</p>
<p class="center">\[
\alpha_L : G \to G,~ (g : o_i \to o_j) \mapsto (g_i^{-1}gg_j : o_i \to o_j).
\]</p>
<p>If <span class="SimpleMath">\(g_1 = 1_g\)</span>, then for all <span class="SimpleMath">\(j\)</span> the rays <span class="SimpleMath">\((r_j : o_1 \to o_j)\)</span> are shifted by <span class="SimpleMath">\(g_j\;\)</span>: so they map to <span class="SimpleMath">\((r_jg_j : o_1 \to o_j)\)</span>. So the operation <code class="code">GroupoidAutomorphismByRayShifts</code> is the special case of <code class="code">GroupoidAutomorphismByNtuple</code> when <span class="SimpleMath">\(g_1=1\)</span>.</p>
</li>
</ul>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">perm1 := [-13,-12,-14];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut1 := GroupoidAutomorphismByObjectPerm( Ha4, perm1 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aut1 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ]
ray images: [ (), (), () ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d := Arrow( Ha4, (1,3,4), -12, -13 ); </span>
[(1,3,4) : -12 -> -13]
<span class="GAPprompt">gap></span> <span class="GAPinput">d1 := ImageElm( aut1, d ); </span>
[(1,3,4) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">gensa4 := GeneratorsOfGroup( a4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha2 := GroupHomomorphismByImages( a4, a4, gensa4, [(2,3,4), (1,3,4)] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut2 := GroupoidAutomorphismByGroupAuto( Ha4, alpha2 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aut2 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (2,3,4)
(2,3,4) -> (1,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (), () ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d2 := ImageElm( aut2, d1 );</span>
[(1,2,4) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">L3 := [(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut3 := GroupoidAutomorphismByNtuple( Ha4, L3 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aut3 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,2)
(2,3,4) -> (1,4,3)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (1,4)(2,3), (1,3)(2,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d3 := ImageElm( aut3, d2 );</span>
[(2,3,4) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">L4 := [(), (1,3,2), (2,4,3)];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut4 := GroupoidAutomorphismByRayShifts( Ha4, L4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aut4 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (1,3,2), (2,4,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d4 := ImageElm( aut4, d3 );</span>
[() : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">h4 := Arrow( Ha4, (2,3,4), -12, -13 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut1234 := aut1*aut2*aut3*aut4;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( aut1234 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,3)
(2,3,4) -> (1,2,3)
object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ]
ray images: [ (), (2,3,4), (1,3,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d4 = ImageElm( aut1234, d );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">inv1234 := InverseGeneralMapping( aut1234 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( inv1234 ); </span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,3)
(2,3,4) -> (2,4,3)
object map: [ -14, -13, -12 ] -> [ -12, -14, -13 ]
ray images: [ (), (1,3,2), (1,3,4) ]
</pre></div>
<p><a id="X86BD61567B41B139" name="X86BD61567B41B139"></a></p>
<h5>6.1-2 GroupoidInnerAutomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupoidInnerAutomorphism</code>( <var class="Arg">gpd</var>, <var class="Arg">arrow</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">&#an style='color: green'>8227; GroupoidInnerAutomorphismNormalSubgroupoid</code>( <var class="Arg">gpd</var>, <var class="Arg">subgpd</var>, <var class="Arg">arrow</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Given an arrow <span class="SimpleMath">\(a = (c : p \to q) \in G\)</span> with <span class="SimpleMath">\(p \neq q\)</span>, the <em>inner automorphism</em> <span class="SimpleMath">\(\alpha_a\)</span> of <span class="SimpleMath">\(G\)</span> by <span class="SimpleMath">\(a\)</span> is the mapping <span class="SimpleMath">\(g \mapsto g^a\)</span> where conjugation of arrows is defined in section <a href="chap4_mj.html#X8653FC9786E3209A"><span class="RefLink">4.5</span></a>. It is easily checked that if <span class="SimpleMath">\(L_a = [1,\ldots,1,c^{-1},1,\ldots,1,c,1,\ldots,1]\)</span>, with <span class="SimpleMath">\(c^{-1}\)</span> in position <span class="SimpleMath">\(p\)</span> and <span class="SimpleMath">\(c\)</span> in position <span class="SimpleMath">\(q\)</span>, then</p>
<p class="center">\[
\alpha_a ~=~ \alpha_{(p,q)} * \alpha_{L_a}.
\]</p>
<p>Similarly, when <span class="SimpleMath">\(p=q\)</span>, then <span class="SimpleMath">\(\alpha_a = \alpha_{L_a}\)</span> where now <span class="SimpleMath">\( L_a = [1,\ldots,1,c,1,\ldots,1]\)</span>, with <span class="SimpleMath">\(c\)</span> in position <span class="SimpleMath">\(p\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">inn1 := GroupoidInnerAutomorphism( Ha4, h4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( inn1 );</span>
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -12, -13 ]
ray images: [ (), (2,4,3), (2,3,4) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">d5 := ImageElm( inn1, d4 );</span>
[(2,3,4) : -14 -> -13]
</pre></div>
<p>Conjugation may also be applied to certain normal subgroupoids of <span class="SimpleMath">\(G\)</span>. Firstly, let <span class="SimpleMath">\(N\)</span> be the wide subgroupoid of <span class="SimpleMath">\(G\)</span> determined by a normal subgroup <span class="SimpleMath">\(n\)</span> of the root group. Then, provided the group element of <span class="SimpleMath">\(a\)</span> is in <span class="SimpleMath">\(n\)</span>, the inner automorphism by <span class="SimpleMath">\(a\)</span> may be applied to <span class="SimpleMath">\(N\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Nk4 := SubgroupoidBySubgroup( Ha4, k4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Nk4, "Nk4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e4 := Arrow( Ha4, (1,2)(3,4), -14, -13 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inn2 := GroupoidInnerAutomorphismNormalSubgroupoid( Ha4, Nk4, e4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( inn2 );</span>
homomorphism to single piece groupoid: Nk4 -> Nk4
root group homomorphism:
(1,2)(3,4) -> (1,2)(3,4)
(1,3)(2,4) -> (1,3)(2,4)
object map: [ -14, -13, -12 ] -> [ -13, -14, -12 ]
ray images: [ (), (), (1,2)(3,4) ]
</pre></div>
<p>Secondly, if <span class="SimpleMath">\(H\)</span> is a homogeneous, discrete subgroupoid of <span class="SimpleMath">\(G\)</span> and if the group element of <span class="SimpleMath">\(a\)</span> is in the common vertex groups, then the inner automorphism may be applied to <span class="SimpleMath">\(H\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Ma4 := MaximalDiscreteSubgroupoid( Ha4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Ma4, "Ma4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inn3 := GroupoidInnerAutomorphism( Ha4, Ma4, e4 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( inn3 );</span>
homogeneous discrete groupoid mapping: [ Ma4 ] -> [ Ma4 ]
images of objects: [ -13, -14, -12 ]
object homomorphisms:
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] )
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] )
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] )
</pre></div>
<p><a id="X8278B93E7F78038C" name="X8278B93E7F78038C"></a></p>
<h5>6.1-3 <span class="Heading">Automorphisms of a groupoid with rays</span></h5>
<p>Let <span class="SimpleMath">\(S\)</span> be a wide subgroupoid with rays of a standard groupoid <span class="SimpleMath">\(G\)</span>.</p>
<p>An automorphism <span class="SimpleMath">\(\alpha\)</span> of the root group <span class="SimpleMath">\(H\)</span> extends to the whole of <span class="SimpleMath">\(S\)</span> with the rays fixed by the automorphism: <span class="SimpleMath">\((r^{-1}_ihr_j : o_i \to o_j) \mapsto (r^{-1}_i (\alpha h)r_j : o_i \to o_j)\)</span>.</p>
<p>An automorphism of <span class="SimpleMath">\(G\)</span> obtained by permuting the objects may map <span class="SimpleMath">\(S\)</span> to a different subgroupoid. So we construct an isomorphism <span class="SimpleMath">\(\iota\)</span> from <span class="SimpleMath">\(S\)</span> to a standard groupoid <span class="SimpleMath">\(T\)</span>, construct <span class="SimpleMath">\(\alpha\)</span> permuting the objects of <span class="SimpleMath">\(T\)</span>, and return <span class="SimpleMath">\(\iota*\alpha*\iota^{-1}\)</span>.</p>
<p>For an automorphism by ray shifts we require that the shifts are elements of the root group of <span class="SimpleMath">\(S\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">## (1) automorphism by group auto </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a6 := GroupHomomorphismByImages( k4, k4,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (1,2)(3,4), (1,3)(2,4) ], [ (1,3)(2,4), (1,4)(2,3) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut6 := GroupoidAutomorphismByGroupAuto( Kk4, a6 );</span>
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
[ [(1,3)(2,4) : -14 -> -14], [(1,4)(2,3) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">a := Arrow( Kk4, (1,3)(2,4), -12, -12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut6, a );</span>
[(1,4)(2,3) : -12 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">b := Arrow( Kk4, (1,4,2), -12, -13 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut6, b );</span>
[(1,2,3) : -12 -> -13]
<span class="GAPprompt">gap></span> <span class="GAPinput">## (2) automorphism by object perm </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut7 := GroupoidAutomorphismByObjectPerm( Kk4, [-13,-12,-14] );</span>
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
[ [(1,4)(2,3) : -13 -> -13], [(1,2)(3,4) : -13 -> -13],
[(2,3,4) : -13 -> -12], [(1,4,3) : -13 -> -14] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut7, a ); </span>
[(1,3)(2,4) : -14 -> -14]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut7, b ); </span>
[(1,3)(2,4) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">## (3) automorphism by ray shifts </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut8 := GroupoidAutomorphismByRayShifts( Kk4,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (), (1,4)(2,3), (1,3)(2,4) ] );</span>
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
[ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,2,3) : -14 -> -13], [(1,2)(3,4) : -14 -> -12] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut8, a );</span>
[(1,3)(2,4) : -12 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut8, b );</span>
[(1,2,3) : -12 -> -13]
<span class="GAPprompt">gap></span> <span class="GAPinput">## (4) combine these three automorphisms </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut678 := aut6 * aut7 * aut8;</span>
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
[ [(1,2)(3,4) : -13 -> -13], [(1,3)(2,4) : -13 -> -13],
[(1,4,3) : -13 -> -12], [(1,3,2) : -13 -> -14] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut678, a ); </span>
[(1,4)(2,3) : -14 -> -14]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( aut678, b );</span>
[(1,4)(2,3) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">## (5) conjgation by an arrow</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e8 := Arrow( Kk4, (1,3)(2,4), -14, -12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut9 := GroupoidInnerAutomorphism( Kk4, e8 );</span>
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
[(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
[ [(1,2)(3,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -12],
[(1,4,2) : -12 -> -13], [(1,4)(2,3) : -12 -> -14] ] ]
</pre></div>
<p><a id="X7DB9D5737EB8C260" name="X7DB9D5737EB8C260"></a></p>
<h5>6.1-4 AutomorphismGroupOfGroupoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AutomorphismGroupOfGroupoid</code>( <var class="Arg">gpd</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">&#an style='color: green'>8227; NiceObjectAutoGroupGroupoid</code>( <var class="Arg">gpd</var>, <var class="Arg">aut</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>As above, let <span class="SimpleMath">\(G\)</span> be the direct product of a group <span class="SimpleMath">\(g\)</span> and a complete digraph with <span class="SimpleMath">\(n\)</span> objects. The <code class="code">AutomorphismGroup</code> <span class="SimpleMath">\(\Aut(G)\)</span> of <span class="SimpleMath">\(G\)</span> is isomorphic to the quotient of <span class="SimpleMath">\(S_n \times A \times g^n\)</span> by a subgroup isomorphic to <span class="SimpleMath">\(g\)</span>, where <span class="SimpleMath">\(A\)</span> is the automorphism group of <span class="SimpleMath">\(g\)</span> and <span class="SimpleMath">\(S_n\)</span> is the symmetric group on the <span class="SimpleMath">\(n\)</span> objects. This is one of the main topics in <a href="chapBib_mj.html#biBAlWe">[AW10]</a>.</p>
<p>If <span class="SimpleMath">\(H\)</span> is the union of <span class="SimpleMath">\(k\)</span> groupoids, all isomorphic to <span class="SimpleMath">\(G\)</span>, then <span class="SimpleMath">\(\Aut(H)\)</span> is isomorphic to <span class="SimpleMath">\(S_k \ltimes \Aut(G)\)</span>.</p>
<p>The function <code class="code">NiceObjectAutoGroupGroupoid</code> takes a groupoid and a subgroup of its automorphism group and retuns a <em>nice monomorphism</em> from this automorphism group to a pc-group, if one is available. The current implementation is experimental. Note that <code class="code">ImageElm</code> at present only works on generating elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AHa4 := AutomorphismGroupOfGroupoid( Ha4 ); </span>
Aut(Ha4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Agens := GeneratorsOfGroup( AHa4);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( Agens );</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">NHa4 := NiceObject( AHa4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MHa4 := NiceMonomorphism( AHa4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( AHa4 ); ## (3!)x24x(12^2)</span>
20736
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( AHa4, "AHa4" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( NHa4, "NHa4" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">## either of these names may be returned</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">names := [ "(((A4 x A4 x A4) : C2) : C3) : C2",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "(C2 x C2 x C2 x C2 x C2 x C2) : (((C3 x C3 x C3) : C3) : (C2 x C2))" ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( NHa4 ) in names;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">## cannot test images of Agens because of random variations </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">## Now do some tests!</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mgi := MappingGeneratorsImages( MHa4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">autgen := mgi[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pcgen := mgi[2];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ngen := Length( autgen );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( [1..ngen], i -> Order(autgen[i]) = Order(pcgen[i]) ); </span>
true
</pre></div>
<p><a id="X7EC237ED7E1978B0" name="X7EC237ED7E1978B0"></a></p>
<h5>6.1-5 <span class="Heading">Inner automorphisms</span></h5>
<p>The inner automorphism subgroup <span class="SimpleMath">\(\mathrm{Inn}(G)\)</span> of the automorphism group of <span class="SimpleMath">\(G\)</span> is the group of inner automorphisms <span class="SimpleMath">\(\wedge a : b \mapsto b^a\)</span> for <span class="SimpleMath">\(a \in G\)</span>. It is <em>not</em> the case that the map <span class="SimpleMath">\(G \to \mathrm{Inn}(G), a \mapsto \wedge a\)</span> preserves multiplication. Indeed, when <span class="SimpleMath">\(a=(o,g,p), b=(p,h,r) \in G\)</span> with objects <span class="SimpleMath">\(p,q,r\)</span> all distict, then</p>
<p class="center">\[
\wedge(ab) ~=~ (\wedge a)(\wedge b)(\wedge a)
~=~ (\wedge b)(\wedge a)(\wedge b).
\]</p>
<p>(Compare this with the permutation identity <span class="SimpleMath">\((pq)(qr)(pq) = (pr) = (qr)(pq)(qr)\)</span>.) So the map <span class="SimpleMath">\(G \to \mathrm{Inn}(G)\)</span> is of type <code class="code">IsMappingWithObjectsByFunction</code>.</p>
<p>In the example we convert the automorphism group <code class="code">AGa4</code> into a single object groupoid, and then define the inner automorphism map.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AHa40 := Groupoid( AHa4, [0] );</span>
single piece groupoid: < Aut(Ha4), [ 0 ] >
<span class="GAPprompt">gap></span> <span class="GAPinput">conj := function(a) </span>
<span class="GAPprompt">></span> <span class="GAPinput"> return ArrowNC( Ha4, true, GroupoidInnerAutomorphism(Ha4,a), 0, 0 ); </span>
<span class="GAPprompt">></span> <span class="GAPinput"> end;; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inner := MappingWithObjectsByFunction( Ha4, AHa40, conj, [0,0,0] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a1 := Arrow( Ha4, (1,2,3), -14, -13 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inner1 := ImageElm( inner, a1 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a2 := Arrow( Ha4, (2,3,4), -13, -12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inner2 := ImageElm( inner, a2 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a3 := a1*a2; </span>
[(1,3)(2,4) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">inner3 := ImageElm( inner, a3 ); </span>
[groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
[() : -14 -> -12] ],
[ [(1,3,4) : -12 -> -12], [(1,2,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -13],
[() : -12 -> -14] ] ] : 0 -> 0]
<span class="GAPprompt">gap></span> <span class="GAPinput">(inner3 = inner1*inner2*inner1) and (inner3 = inner2*inner1*inner2);</span>
true
true
</pre></div>
<p><a id="X80F9594481CA9FDE" name="X80F9594481CA9FDE"></a></p>
<h5>6.1-6 GroupoidAutomorphismByGroupAutos</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupoidAutomorphismByGroupAutos</code>( <var class="Arg">gpd</var>, <var class="Arg">auts</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Homogeneous, discrete groupoids are the second type of groupoid for which a method is provided for <code class="code">AutomorphismGroupOfGroupoid</code>. This is used in the <strong class="pkg">XMod</strong> package for constructing crossed modules of groupoids. The two types of generating automorphism are <code class="code">GroupoidAutomorphismByGroupAutos</code>, which requires a list of group automorphisms, one for each object group, and <code class="code">GroupoidAutomorphismByObjectPerm</code>, which permutes the objects. So, if the object groups <span class="SimpleMath">\(g\)</span> have automorphism group <span class="SimpleMath">\(\Aut(g)\)</span> and there are <span class="SimpleMath">\(n\)</span> objects, the autmorphism group of the groupoid has size <span class="SimpleMath">\(n!|\Aut(g)|^n\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dd8 := HomogeneousDiscreteGroupoid( d8, [ -13..-10] ); </span>
homogeneous, discrete groupoid: < d8, [ -13 .. -10 ] >
<span class="GAPprompt">gap></span> <span class="GAPinput">aut10 := GroupoidAutomorphismByObjectPerm( Dd8, [-12,-10,-11,-13] ); </span>
groupoid homomorphism : morphism from a homogeneous discrete groupoid:
[ -13, -12, -11, -10 ] -> [ -12, -10, -11, -13 ]
object homomorphisms:
IdentityMapping( d8 )
IdentityMapping( d8 )
IdentityMapping( d8 )
IdentityMapping( d8 )
<span class="GAPprompt">gap></span> <span class="GAPinput">gend8 := GeneratorsOfGroup( d8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g1 := gend8[1];; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g2 := gend8[2];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b1 := IdentityMapping( d8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b2 := GroupHomomorphismByImages( d8, d8, gend8, [g1, g2*g1 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b3 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b4 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2^(g1*g2) ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut11 := GroupoidAutomorphismByGroupAutos( Dd8, [b1,b2,b3,b4] ); </span>
groupoid homomorphism : morphism from a homogeneous discrete groupoid:
[ -13, -12, -11, -10 ] -> [ -13, -12, -11, -10 ]
object homomorphisms:
IdentityMapping( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ],
[ (5,6,7,8), (5,8)(6,7) ] )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (6,8) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">ADd8 := AutomorphismGroupOfGroupoid( Dd8 ); </span>
<group with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( ADd8 ); ## 4!*8^4</span>
98304
<span class="GAPprompt">gap></span> <span class="GAPinput">genADd8 := GeneratorsOfGroup( ADd8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( genADd8 ); </span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">w := GroupoidAutomorphismByGroupAutos( Dd8, [b2,b1,b1,b1] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := GroupoidAutomorphismByGroupAutos( Dd8, [b3,b1,b1,b1] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">y := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -11, -10, -13 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">z := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -13, -11, -10 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ok := ForAll( genADd8, a -> a in[ w, x, y, z ] ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NADd8 := NiceObject( ADd8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MADd8 := NiceMonomorphism( ADd8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">w1 := ImageElm( MADd8, w );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x1 := ImageElm( MADd8, x );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">y1 := ImageElm( MADd8, y );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">z1 := ImageElm( MADd8, z );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u := z*w*y*x*z; </span>
groupoid homomorphism : morphism from a homogeneous discrete groupoid:
[ -13, -12, -11, -10 ] -> [ -11, -13, -10, -12 ]
object homomorphisms:
IdentityMapping( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ],
[ (5,6,7,8), (5,8)(6,7) ] )
IdentityMapping( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">u1 := z1*w1*y1*x1*z1; </span>
(1,2,4,3)(5,17,23,11,6,18,24,16)(7,19,25,15,9,21,27,13)(8,20,26,14,10,22,28,12)
<span class="GAPprompt">gap></span> <span class="GAPinput">imu := ImageElm( MADd8, u );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">u1 = imu;</span>
true
</pre></div>
<p><a id="X7A93C6507BC697CD" name="X7A93C6507BC697CD"></a></p>
<h5>6.1-7 AutomorphismGroupoidOfGroupoid</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AutomorphismGroupoidOfGroupoid</code>( <var class="Arg">gpd</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <span class="SimpleMath">\(G\)</span> is a single piece groupoid with automorphism group <span class="SimpleMath">\(\Aut(G)\)</span>, and if <span class="SimpleMath">\(H\)</span> is the union of <span class="SimpleMath">\(k\)</span> pieces, all isomorphic to <span class="SimpleMath">\(G\)</span>, then the automorphism group of <span class="SimpleMath">\(H\)</span> is the wreath product <span class="SimpleMath">\(S_k \ltimes \Aut(G)\)</span>. However, we find it more convenient to construct the <em>automorphism groupoid</em> of <span class="SimpleMath">\(H\)</span>. This is a single piece groupoid <span class="SimpleMath">\(\AUT(H)\)</span> with <span class="SimpleMath">\(k\)</span> objects - the object lists of the pieces of <span class="SimpleMath">\(H\)</span> - and root group <span class="SimpleMath">\(\Aut(G)\)</span>. Isomorphisms between the root groups of the <span class="SimpleMath">\(k\)</span> pieces may be applied to the generators of <span class="SimpleMath">\(\Aut(G)\)</span> to construct automorphism groups of these pieces, and then isomorphisms between these automorphism groups. We then construct <span class="SimpleMath">\(\AUT(H)\)</span> using <code class="code">GroupoidByIsomorphisms</code>.</p>
<p>In the special case that <span class="SimpleMath">\(H\)</span> is homogeneous, there is no need to construct a collection of automorphism groups. Rather, the rays of <span class="SimpleMath">\(\AUT(H)\)</span> are given by <code class="code">IsomorphismNewObjects</code>. For the example we use <code class="code">HGd8</code> constructed in subsection <code class="func">HomogeneousGroupoid</code> (<a href="chap4_mj.html#X855F318181808814"><span class="RefLink">4.1-5</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">HGd8 := HomogeneousGroupoid( Gd8, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ [-39,-38,-37], [-36,-35,-34], [-33,-32,-31] ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( HGd8, "HGd8" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AHGd8 := AutomorphismGroupoidOfGroupoid( HGd8 ); </span>
Aut(HGd8)
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjectList( AHGd8 );</span>
[ [ -39, -38, -37 ], [ -36, -35, -34 ], [ -33, -32, -31 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RaysOfGroupoid( AHGd8 ){[2..3]};</span>
[ groupoid homomorphism :
[ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38],
[() : -39 -> -37] ],
[ [(5,6,7,8) : -36 -> -36], [(5,7) : -36 -> -36], [() : -36 -> -35],
[() : -36 -> -34] ] ], groupoid homomorphism :
[ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38],
[() : -39 -> -37] ],
[ [(5,6,7,8) : -33 -> -33], [(5,7) : -33 -> -33], [() : -33 -> -32],
[() : -33 -> -31] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">obgp := ObjectGroup( AHGd8, [ -36, -35, -34 ] );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( obgp ); ## 3!*8^3</span>
3072
</pre></div>
<p><a id="X8003AF117B956D16" name="X8003AF117B956D16"></a></p>
<h4>6.2 <span class="Heading">Matrix representations of groupoids</span></h4>
<p>Suppose that <code class="code">gpd</code> is the direct product of a group <span class="SimpleMath">\(G\)</span> and a complete digraph, and that <span class="SimpleMath">\(\rho : G \to M\)</span> is an isomorphism to a matrix group <span class="SimpleMath">\(M\)</span>. Then, if <code class="code">rep</code> is the isomorphic groupoid with the same objects and root group <span class="SimpleMath">\(M\)</span>, there is an isomorphism <span class="SimpleMath">\(\mu\)</span> from <code class="code">gpd</code> to <code class="code">rep</code> mapping <span class="SimpleMath">\((g : i \to j)\)</span> to <span class="SimpleMath">\((\rho g : i \to j)\)</span>.</p>
<p>When <code class="code">gpd</code> is a groupoid with rays, a representation can be obtained by restricting a representation of its parent.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">reps := IrreducibleRepresentations( a4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep4 := reps[4]; </span>
Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
[ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ],
[ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
[ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Ra4 := Groupoid( Image( rep4 ), Ga4!.objects );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjectList( Ra4 ) = [ -15 .. -11 ];</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfGroupoid( Ga4 );</span>
[ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14],
[() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">images := List( gens, </span>
<span class="GAPprompt">></span> <span class="GAPinput"> g -> Arrow( Ra4, ImageElm(rep4,g![2]), g![3], g![4] ) ); </span>
[ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15],
[[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -15 -> -15],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">mor := GroupoidHomomorphismFromSinglePiece( Ga4, Ra4, gens, images );
</span>
groupoid homomorphism :
[ [ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14],
[() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ],
[ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15],
[[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -15 -> -15],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMatrixGroupoid( Ra4 ); </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a := Arrow( Ha4, (1,4,2), -12, -13 );</span>
[(1,4,2) : -12 -> -13]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( mor, a );</span>
[[ [ 0, 0, 1 ], [ -1, 0, 0 ], [ 0, -1, 0 ] ] : -12 -> -13]
<span class="GAPprompt">gap></span> <span class="GAPinput">rmor := RestrictedMappingGroupoids( mor, Ha4 );</span>
groupoid homomorphism :
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
[() : -14 -> -12] ],
[ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -14 -> -14],
[[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -14 -> -14],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -13],
[[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -12] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ParentMappingGroupoids( rmor ) = mor;</span>
true
</pre></div>
<p><a id="X7DB8E7EF7A51F1BE" name="X7DB8E7EF7A51F1BE"></a></p>
<h4>6.3 <span class="Heading">Groupoid actions</span></h4>
<p>Recall from sections <a href="chap4_mj.html#X8653FC9786E3209A"><span class="RefLink">4.5</span></a> and <code class="func">GroupoidInnerAutomorphism</code> (<a href="chap6_mj.html#X86BD61567B41B139"><span class="RefLink">6.1-2</span></a>) the notion of <em>conjugation</em> in a groupoid, and the associated inner automorphisms.</p>
<p>It was mentioned there that the map <span class="SimpleMath">\(\wedge : G \to \Aut(G),~ a \to \wedge a\)</span>, is <em>not</em> a groupoid homomorphism. It is in fact a <em>groupoid action</em> which we now define. Let <span class="SimpleMath">\(\{p,q,r,u,v\}\)</span> be distinct objects in <span class="SimpleMath">\(G\)</span> and let:</p>
<p><span class="SimpleMath">\(a_1 = (c_1 : p \to q),~~ a_2 = (c_2 : q \to r),~~ a_3 = (c_3 : q \to p),~~ a_4 = (c_4 : u \to v)\)</span>,</p>
<p><span class="SimpleMath">\(b_1 = (d_1 : p \to p),~~ b_2 = (d_2 : p \to p),~~ b_3 = (d_3 : q \to q),~~ b_4 = (c_3c_1 : q \to q)\)</span> be arrows in <span class="SimpleMath">\(G\)</span>. Then the following <em>conjugation identities</em> must be satisfied:</p>
<ul>
<li><p><span class="SimpleMath">\(\wedge(a_1a_2) ~=~ (\wedge a_1)*(\wedge a_2)*(\wedge a_1) ~=~ (\wedge a_2)*(\wedge a_1)*(\wedge a_2)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(\wedge(b_1a_1) ~=~ (\wedge b_1)*(\wedge a_1)*(\wedge b_1)^{-1}\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(\wedge(a_1b_3) ~=~ (\wedge b_3)^{-1}*(\wedge a_1)*(\wedge b_3)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(\wedge(b_1b_2) ~=~ (\wedge b_1)*(\wedge b_2)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\(\wedge(a_1a_3) ~=~ (\wedge a_1)*(\wedge a_3)*(\wedge b_4)\)</span>,</p>
</li>
<li><p><span class="SimpleMath">\((\wedge a_1)*(\wedge a_4) ~=~ (\wedge a_4)*(\wedge a_1)\)</span>.</p>
</li>
</ul>
<p>In the following example we check the first of these identities in one particular case.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c1 := Arrow( Ha4, (1,2)(3,4), -14, -13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">innc1 := GroupoidInnerAutomorphism( Ha4, c1 );</span>
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
[() : -14 -> -12] ],
[ [(1,4,2) : -13 -> -13], [(1,4,3) : -13 -> -13], [() : -13 -> -14],
[(1,2)(3,4) : -13 -> -12] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c2 := Arrow( Ha4, (1,4,2), -13, -12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">innc2 := GroupoidInnerAutomorphism( Ha4, c2 );</span>
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
[() : -14 -> -12] ],
[ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [(1,4,2) : -14 -> -12],
[(1,2,4) : -14 -> -13] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">c12 := c1 * c2;</span>
[(2,4,3) : -14 -> -12]
<span class="GAPprompt">gap></span> <span class="GAPinput">innc12 := GroupoidInnerAutomorphism( Ha4, c12 );</span>
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
[() : -14 -> -12] ],
[ [(1,4,2) : -12 -> -12], [(2,3,4) : -12 -> -12], [(2,3,4) : -12 -> -13],
[(2,4,3) : -12 -> -14] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">[ innc1 * innc2 * innc1 = innc12, innc2 * innc1 * innc2 = innc12 ];</span>
[ true, true ]
</pre></div>
<p><a id="X7C4DCB287B3DD0CD" name="X7C4DCB287B3DD0CD"></a></p>
<h5>6.3-1 GroupoidActionByConjugation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GroupoidActionByConjugation</code>( <var class="Arg">gpd</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">&#an style='color: green'>8227; IsGroupoidAction</code>( <var class="Arg">map</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ActionMap</code>( <var class="Arg">act</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The operation <code class="code">GroupoidInnerAutomorphism</code>, which produces the conjugation action of <span class="SimpleMath">\(G\)</span> on itself, does satisfy the conjugation identities and so provides a standard example of an action.</p>
<p>An action is a record with fields <code class="code">Source</code>, <code class="code">Range</code> and <code class="code">ActionMap</code>.</p>
<p>The examples repeat those in section <code class="func">GroupoidInnerAutomorphism</code> (<a href="chap6_mj.html#X86BD61567B41B139"><span class="RefLink">6.1-2</span></a>): firstly with a groupoid acting on itself.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">act1 := GroupoidActionByConjugation( Ha4 );</span>
<general mapping: Ha4 -> Aut(Ha4) >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupoidAction( act1 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">amap1 := ActionMap( act1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">amap1( h4 ) = inn1;</span>
true
</pre></div>
<p>Secondly with an action on a single piece, normal subgroupoid.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">act2 := GroupoidActionByConjugation( Ha4, Nk4 );</span>
<general mapping: Ha4 -> Aut(Nk4) >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupoidAction( act2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">amap2 := ActionMap( act2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">amap2( e4 ) = inn2;</span>
true
</pre></div>
<p>Thirly with an action on a homogeneous, discrete subgroupoid.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">act3 := GroupoidActionByConjugation( Ha4, Ma4 );</span>
<general mapping: Ha4 -> Aut(Ma4) >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGroupoidAction( act3 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">amap3 := ActionMap( act3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">amap3( e4 ) = inn3;</span>
true
</pre></div>
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