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

Synopsis

Description

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

                     R[p , p , p ]
                        0   1   2
o3 = ---------------------------------------------
     (- b*p  + a*p , - c*p  + b*p , - c*p  + a*p )
           0      1       1      2       0      2

o3 : QuotientRing
i4 : symmetricAlgebra(R^{1,2,3})

o4 = R[p , p , p ]
        0   1   2

o4 : PolynomialRing

Most of the optional arguments for monoids (see Ring Array or monoid) are available here as well, as in the following example. They apply to the variables that correspond to the generators of the module.

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

                    R[x, y, z]
o5 = ---------------------------------------
     (- b*x + a*y, - c*y + b*z, - c*x + a*z)

o5 : QuotientRing
i6 : symmetricAlgebra(M, VariableBaseName=>G, MonomialSize=>16)

                     R[G , G , G ]
                        0   1   2
o6 = ---------------------------------------------
     (- b*G  + a*G , - c*G  + b*G , - c*G  + a*G )
           0      1       1      2       0      2

o6 : QuotientRing
i7 : symmetricAlgebra(M, Degrees=> {3:1})

                     R[p , p , p ]
                        0   1   2
o7 = ---------------------------------------------
     (- b*p  + a*p , - c*p  + b*p , - c*p  + a*p )
           0      1       1      2       0      2

o7 : QuotientRing

Caveat

This function predates the ability to create polynomial rings over polynomial rings, and perhaps that is what should be returned.