# symmetricPower(ZZ,LabeledModule) -- Symmetric power of a @TO LabeledModule@

## Description

This produces the symmetric 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 multiSubsets(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 : F=symmetricPower(2,M) 6 o3 = S o3 : free S-module with labeled basis i4 : basisList F o4 = {{0, 0}, {0, 1}, {1, 1}, {0, 2}, {1, 2}, {2, 2}} o4 : List i5 : underlyingModules F 3 o5 = {S } o5 : List i6 : G=symmetricPower(2,F); o6 : free S-module with labeled basis i7 : basisList G o7 = {{{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}, {{0, 1}, {0, 1}}, {{0, 0}, {1, 1}}, ------------------------------------------------------------------------ {{0, 1}, {1, 1}}, {{1, 1}, {1, 1}}, {{0, 0}, {0, 2}}, {{0, 1}, {0, 2}}, ------------------------------------------------------------------------ {{1, 1}, {0, 2}}, {{0, 2}, {0, 2}}, {{0, 0}, {1, 2}}, {{0, 1}, {1, 2}}, ------------------------------------------------------------------------ {{1, 1}, {1, 2}}, {{0, 2}, {1, 2}}, {{1, 2}, {1, 2}}, {{0, 0}, {2, 2}}, ------------------------------------------------------------------------ {{0, 1}, {2, 2}}, {{1, 1}, {2, 2}}, {{0, 2}, {2, 2}}, {{1, 2}, {2, 2}}, ------------------------------------------------------------------------ {{2, 2}, {2, 2}}} o7 : List

The first symmetric 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 $symmetricPower(1,M)$ 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=symmetricPower(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=symmetricPower(0,M) 1 o17 = S o17 : free S-module with labeled basis i18 : basisList E o18 = {{}} o18 : List i19 : underlyingModules E o19 = {} o19 : List