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]; |
i2 : M=labeledModule(S^3); o2 : free S-module with labeled basis |
i3 : E=exteriorPower(2,M)
3
o3 = S
o3 : free S-module with labeled basis
|
i4 : basisList E
o4 = {{0, 1}, {0, 2}, {1, 2}}
o4 : List
|
i5 : underlyingModules E
3
o5 = {S }
o5 : List
|
i6 : F=exteriorPower(2,E); o6 : free S-module with labeled basis |
i7 : basisList F
o7 = {{{0, 1}, {0, 2}}, {{0, 1}, {1, 2}}, {{0, 2}, {1, 2}}}
o7 : List
|
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]; |
i9 : M=labeledModule(S^2); o9 : free S-module with labeled basis |
i10 : E=exteriorPower(1,M); o10 : free S-module with labeled basis |
i11 : basisList M
o11 = {0, 1}
o11 : List
|
i12 : basisList E
o12 = {{0}, {1}}
o12 : List
|
i13 : underlyingModules M
o13 = {}
o13 : List
|
i14 : underlyingModules E
2
o14 = {S }
o14 : List
|
By convention, the zeroeth symmetric power of an $S$-module is the labeled module $S^1$ with basis list $\{\{\}\}$ and with no underlying modules.
i15 : S=ZZ/101[x,y,z]; |
i16 : M=labeledModule(S^2); o16 : free S-module with labeled basis |
i17 : E=exteriorPower(0,M)
1
o17 = S
o17 : free S-module with labeled basis
|
i18 : basisList E
o18 = {{}}
o18 : List
|
i19 : underlyingModules E
o19 = {}
o19 : List
|