symmetricAlgebra(Module) -- the symmetric algebra of a module

i1 : R = QQ[a..d];

i2 : M = image matrix{{a,b,c}}

o2 = image | a b c |

                             1
o2 : R-module, submodule of R

i3 : symmetricAlgebra M

         QQ [p , p , p , p , p , p , p ]
              0   1   2   3   4   5   6
o3 = ---------------------------------------
     (p p  - p p , p p  - p p , p p  - p p )
       1 3    0 4   2 4    1 5   2 3    0 5

o3 : QuotientRing

i4 : symmetricAlgebra(R^{1,2,3})

o4 = QQ [p , p , p , p , p , p , p ]
          0   1   2   3   4   5   6

o4 : PolynomialRing

i5 : symmetricAlgebra(M, Variables=>{x,y,z})

         QQ [p , p , p , p , p , p , p ]
              0   1   2   3   4   5   6
o5 = ---------------------------------------
     (p p  - p p , p p  - p p , p p  - p p )
       1 3    0 4   2 4    1 5   2 3    0 5

o5 : QuotientRing

i6 : symmetricAlgebra(M, VariableBaseName=>G, MonomialSize=>16)

         QQ [G , G , G , G , G , G , G ]
              0   1   2   3   4   5   6
o6 = ---------------------------------------
     (G G  - G G , G G  - G G , G G  - G G )
       1 3    0 4   2 4    1 5   2 3    0 5

o6 : QuotientRing

i7 : symmetricAlgebra(M, Degrees=> {7:1})

         QQ [p , p , p , p , p , p , p ]
              0   1   2   3   4   5   6
o7 = ---------------------------------------
     (p p  - p p , p p  - p p , p p  - p p )
       1 3    0 4   2 4    1 5   2 3    0 5

o7 : QuotientRing