exteriorPower(ZZ,LabeledModule) -- Exterior power of a @TO LabeledModule@

Synopsis

Function: exteriorPower
Usage:

E = exteriorPower(i,M)
Inputs:
- i, an integer,
- M, a free module with labeled basis,
Optional inputs:
- Strategy => ..., default value null,
Outputs:
- a free module with labeled basis,

Description

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

Ways to use this method: