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# ALCO, chapter 5
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been generated by AutoDoc. It contains examples extracted from
# the package documentation. Each example is preceded by a comment which gives
# the name of a GAPDoc XML file and a line range from which the example were
# taken. Note that the XML file in turn may have been generated by AutoDoc
# from some other input.
#
gap> START_TEST("alco05.tst");
# doc/ALCO.xml:1835-1840
gap> M := MOGLeechLatticeGeneratorMatrix;;
gap> G := M*TransposedMat(M)/8;;
gap> G = MOGLeechLatticeGramMatrix;
true
gap> IsLeechLatticeGramMatrix(G);
true
# doc/ALCO.xml:1905-1939
gap> O := OctavianIntegers;;
gap> gens := Concatenation(List(IdentityMat(3), x -> List(Basis(O), y -> x*y)));
[ [ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1, 0*e1 ],
[ (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e1+(-1/2)*e3+(1/2)*e4+(1/2)*e5, 0*e1, 0*e1 ],
[ (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e2+(-1/2)*e4+(1/2)*e5+(-1/2)*e6, 0*e1, 0*e1 ],
[ (-1/2)*e1+(-1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1, 0*e1 ],
[ (1/2)*e1+(-1/2)*e4+(1/2)*e6+(-1/2)*e8, 0*e1, 0*e1 ],
[ 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1 ],
[ 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e1+(-1/2)*e3+(1/2)*e4+(1/2)*e5, 0*e1 ],
[ 0*e1, (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e2+(-1/2)*e4+(1/2)*e5+(-1/2)*e6, 0*e1 ],
[ 0*e1, (-1/2)*e1+(-1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1 ],
[ 0*e1, (1/2)*e1+(-1/2)*e4+(1/2)*e6+(-1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7 ],
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(-1/2)*e3+(1/2)*e4+(1/2)*e5 ],
[ 0*e1, 0*e1, (-1/2)*e2+(1/2)*e3+(-1/2)*e5+(1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(-1/2)*e4+(1/2)*e5+(-1/2)*e6 ],
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e3+(1/2)*e4+(-1/2)*e5 ],
[ 0*e1, 0*e1, (1/2)*e1+(-1/2)*e4+(1/2)*e6+(-1/2)*e8 ] ]
gap> G := IdentityMat(3)*One(O)/2;;
gap> B := CanonicalBasis(O);;
gap> O3 := OctonionLatticeByGenerators(gens, G, B);
<free left module over Integers, with 24 generators>
gap> KnownAttributesOfObject(O3);
[ "LeftActingDomain", "Dimension", "GeneratorsOfLeftOperatorAdditiveGroup",
"UnderlyingOctonionRing", "UnderlyingOctonionRingBasis",
"OctonionGramMatrix", "GeneratorsAsCoefficients",
"LLLReducedBasisCoefficients" ]
# doc/ALCO.xml:1962-1963
gap> UnderlyingOctonionRing(O3);
<algebra-with-one of dimension 8 over Rationals>
# doc/ALCO.xml:1976-1979
gap> OctonionGramMatrix(O3);; Display(last);
[ [ (1/2)*e8, 0*e1, 0*e1 ],
[ 0*e1, (1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (1/2)*e8 ] ]
# doc/ALCO.xml:1992-2016
gap> GeneratorsAsCoefficients(O3);
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ] ]
# doc/ALCO.xml:2027-2051
gap> LLLReducedBasisCoefficients(O3);
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1 ] ]
# doc/ALCO.xml:2063-2066
gap> Dimension(O3);
24
gap> Rank(O3);
24
# doc/ALCO.xml:2077-2081
gap> g := 2*GramMatrix(O3);;
gap> DimensionsMat(g);
[ 24, 24 ]
gap> DeterminantMat(g);
1
# doc/ALCO.xml:2100-2128
gap> IsOctonionLatticeBasis(Basis(O3));
true
gap> b := BasisVectors(Basis(O3));
[ [ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1, 0*e1 ],
[ (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e2+(1/2)*e3+(-1/2)*e6, 0*e1, 0*e1 ],
[ (1/2)*e2+(1/2)*e4+(-1/2)*e5+(-1/2)*e6, 0*e1, 0*e1 ],
[ (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8, 0*e1, 0*e1 ],
[ 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1 ],
[ 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7, 0*e1 ],
[ 0*e1, (1/2)*e1+(1/2)*e3+(1/2)*e4+(-1/2)*e5, 0*e1 ],
[ 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e3+(-1/2)*e6, 0*e1 ],
[ 0*e1, (1/2)*e2+(1/2)*e4+(-1/2)*e5+(-1/2)*e6, 0*e1 ],
[ 0*e1, (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8, 0*e1 ],
[ 0*e1, 0*e1, (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7 ],
[ 0*e1, 0*e1, (-1/2)*e1+(-1/2)*e2+(-1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e4+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-1/2)*e7 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e3+(1/2)*e4+(-1/2)*e5 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e2+(1/2)*e3+(-1/2)*e6 ],
[ 0*e1, 0*e1, (1/2)*e2+(1/2)*e4+(-1/2)*e5+(-1/2)*e6 ],
[ 0*e1, 0*e1, (1/2)*e1+(1/2)*e2+(-1/2)*e5+(-1/2)*e8 ] ]
gap> GramMatrix(O3) = List(b, x -> List(b, y -> ScalarProduct(O3, x, y)));
true
# doc/ALCO.xml:2161-2166
gap> x := Sum(BasisVectors(Basis(O3)){[2,3,4]});
[ (1/2)*e2+(1/2)*e3+(-1/2)*e5+(-3/2)*e7, 0*e1, 0*e1 ]
gap> x in O3;
true
gap> \in( x, O3);
true
# doc/ALCO.xml:2184-2193
gap> b := BasisVectors(Basis(O3));;
gap> b[1];
[ (-1/2)*e1+(1/2)*e5+(1/2)*e6+(1/2)*e7, 0*e1, 0*e1 ]
gap> GramMatrix(O3) = List(b, x -> List(b, y -> ScalarProduct(O3, x, y)));
true
gap> c := LLLReducedBasisCoefficients(O3);;
gap> c[1];
[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> GramMatrix(O3) = List(c, x -> List(c, y -> ScalarProduct(O3, x, y)));
true
# doc/ALCO.xml:2213-2226
gap> s := LinearCombination(OctonionE8Basis, [ 1, 2, 1, 2, 2, 2, 2, 1 ]);;
gap> leech_gens := List(Basis(OctavianIntegers), x ->
> x*[[s,s,0],[0,s,s],ComplexConjugate([s,s,s])]);;
gap> leech_gens := Concatenation(leech_gens);;
gap> Leech := OctonionLatticeByGenerators(leech_gens, G, B);
<free left module over Integers, with 24 generators>
gap> IsLeechLatticeGramMatrix(GramMatrix(Leech));
true
gap> IsSublattice(Leech, O3);
false
gap> IsSublattice(O3, Leech);
true
gap> IsSubset(O3, Leech);
true
# doc/ALCO.xml:2244-2247
gap> rand := Random(O3);;
gap> coeffs := Coefficients(Basis(O3), rand);;
gap> rand = LinearCombination(Basis(O3), coeffs);
true
#
gap> STOP_TEST("alco05.tst", 1);
