This produces the exterior power of a labeled module as a labeled module with the natural basis list. For instance if M is a labeled module with basis list L, then exteriorPower(2,M) is a labeled module with basis list subsets(2,L) and with M as an underlying module,
i1 : S=ZZ/101[x,y,z];
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i2 : M=labeledModule(S^3);
o2 : free S-module with labeled basis
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i3 : E=exteriorPower(2,M)
3
o3 = S
o3 : free S-module with labeled basis
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i4 : basisList E
o4 = {{0, 1}, {0, 2}, {1, 2}}
o4 : List
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i5 : underlyingModules E
3
o5 = {S }
o5 : List
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i6 : F=exteriorPower(2,E);
o6 : free S-module with labeled basis
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i7 : basisList F
o7 = {{{0, 1}, {0, 2}}, {{0, 1}, {1, 2}}, {{0, 2}, {1, 2}}}
o7 : List
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The first exterior power of a labeled module is not the identity in the category of labeled modules. For instance, if M is a free labeled module with basis list {0,1} and with no underlying modules, then exteriorPower(1,M) is a labeled module with basis list {{0}, {1},} and with M as an underlying module.
i8 : S=ZZ/101[x,y,z];
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i9 : M=labeledModule(S^2);
o9 : free S-module with labeled basis
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i10 : E=exteriorPower(1,M);
o10 : free S-module with labeled basis
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i11 : basisList M
o11 = {0, 1}
o11 : List
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i12 : basisList E
o12 = {{0}, {1}}
o12 : List
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i13 : underlyingModules M
o13 = {}
o13 : List
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i14 : underlyingModules E
2
o14 = {S }
o14 : List
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By convention, the zeroeth symmetric power of an S-module is the labeled module S1 with basis list {{}} and with no underlying modules.
i15 : S=ZZ/101[x,y,z];
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i16 : M=labeledModule(S^2);
o16 : free S-module with labeled basis
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i17 : E=exteriorPower(0,M)
1
o17 = S
o17 : free S-module with labeled basis
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i18 : basisList E
o18 = {{}}
o18 : List
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i19 : underlyingModules E
o19 = {}
o19 : List
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