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symmetricMultiplication -- creates the symmetric multiplication map

Synopsis

Description

Given a labeled free module $F$, and two nonnegative integers $i$ and $j$, this yields the multiplication map $$ f: S^i(F)\otimes S^j(F)\to S^{i+j}(F). $$ The output map is treated as a map of labeled modules, and the source and target are inherit the natural structure as labeled modules from $F$. For instance, if the basis list of $F$ is $L$, then the basis list of the target of $f$ is the list multiSubsets(i+j,L).

i1 : S=ZZ/101[x,y,z];
 
i2 : F=labeledModule(S^2);

o2 : free S-module with labeled basis
 
i3 : f=symmetricMultiplication(F,2,2)

o3 = | 1 0 0 0 0 0 0 0 0 |
     | 0 1 0 1 0 0 0 0 0 |
     | 0 0 1 0 1 0 1 0 0 |
     | 0 0 0 0 0 1 0 1 0 |
     | 0 0 0 0 0 0 0 0 1 |

             5       9
o3 : Matrix S  <--- S
 
i4 : source f

      9
o4 = S

o4 : free S-module with labeled basis
 
i5 : basisList F

o5 = {0, 1}

o5 : List
 
i6 : basisList source f

o6 = {{{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}},
     ------------------------------------------------------------------------
     {{0, 1}, {0, 1}}, {{0, 1}, {1, 1}}, {{1, 1}, {0, 0}}, {{1, 1}, {0, 1}},
     ------------------------------------------------------------------------
     {{1, 1}, {1, 1}}}

o6 : List
 
i7 : basisList target f

o7 = {{0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 1}, {0, 1, 1, 1}, {1, 1, 1, 1}}

o7 : List

Ways to use symmetricMultiplication :

For the programmer

The object symmetricMultiplication is a method function.