98 lines
3.1 KiB
Plaintext
98 lines
3.1 KiB
Plaintext
# byconj.gi RAQ Implementation of quandles by conjugation
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InstallMethod(ConjugatorFamily, "for a family",
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[IsFamily],
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function(fam)
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local F;
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# Does GAP provide any way to get at the name of a family other than
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# fam!.NAME ?
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F := NewFamily(Concatenation("ConjugatorFamily(", fam!.NAME, ")"),
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IsConjugatorObject);
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F!.ConjType := NewType(F, IsDefaultConjugatorObject);
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return F;
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end);
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ConjugatorType@ := obj -> ConjugatorFamily(FamilyObj(obj))!.ConjType;
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InstallMethod(ConjugatorObj,
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"for a mult element that allows quotients (and should be assoc)",
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[IsMultiplicativeElementWithInverse],
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obj -> Objectify(ConjugatorType@(obj), [Immutable(obj)])
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);
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## Printing and viewing
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InstallMethod(String, "for conjugator objects",
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[IsDefaultConjugatorObject],
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obj -> Concatenation("ConjugatorObj( ", String(obj![1]), " )")
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);
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InstallMethod(ViewString, "for conjugator objects",
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[IsDefaultConjugatorObject],
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obj -> Concatenation("^", ViewString(obj![1]), ":")
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);
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InstallMethod(UnderlyingMultiplicativeElement, "for a conjugator object",
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[IsDefaultConjugatorObject],
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obj -> obj![1]
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);
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InstallMethod( \=, "for two conjugator objects",
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IsIdenticalObj,
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[IsDefaultConjugatorObject, IsDefaultConjugatorObject],
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function(l,r) return l![1] = r![1]; end
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);
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InstallMethod( \<, "for two conjugator objects",
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IsIdenticalObj,
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[IsDefaultConjugatorObject, IsDefaultConjugatorObject],
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function(l,r) return l![1] < r![1]; end
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);
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InstallMethod( \*, "for two conjugator objects",
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IsIdenticalObj,
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[IsDefaultConjugatorObject, IsDefaultConjugatorObject],
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function(l,r) return ConjugatorObj(LeftQuotient(l![1],r![1])*l![1]); end
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);
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InstallOtherMethod(LeftQuotient, "for two conjugator objects",
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IsIdenticalObj,
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[IsDefaultConjugatorObject,IsDefaultConjugatorObject],
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function(l,r) return ConjugatorObj((l![1]*r![1])/l![1]); end
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);
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InstallMethod(ConjugationQuandle, "for a group",
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[IsGroup and IsFinite],
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function(G)
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local fam, elts, Q;
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fam := CollectionsFamily(ConjugatorFamily(ElementsFamily(FamilyObj(G))));
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# Question: how do we feasibly determine a set of generators of
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# Conj(G) from a set of generators of G, so that we can handle infinite
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# conj-quandles here?
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elts := List(Elements(G), g -> ConjugatorObj(g) );
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# What we would like to do is
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# return AsLeftQuandle[NC?](elts);
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# but that's NIY.
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Q := LeftQuandleNC(fam, elts);
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# We know that elts was actually closed under * and LeftQuotient, and
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# since we are in a method only for finite groups, ergo Q is finite:
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SetIsFinite(Q, true);
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return Q;
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end);
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## Methods that are easier in conjugator quandles:
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InstallMethod(GeneratorsOfMagma,
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"for a quandle generated by conjugator objects",
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[IsLeftQuasigroup and HasGeneratorsOfLeftQuasigroup and
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IsConjugatorObjectCollection],
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function(Q)
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local gens, invgens;
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# idea: ^g^-1: * ^h: = ^g: \ ^h:, so the generators of the quasigroup
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# together with their inverses generate the quandle as a magma.
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gens := GeneratorsOfLeftQuasigroup(Q);
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invgens := Set(gens, g -> ConjugatorObj(Inverse(g![1])));
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UniteSet(invgens, gens);
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return invgens;
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end);
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