# symmetricMultiplication -- creates the symmetric multiplication map

## Synopsis

• Usage:
symmetricMultiplication(F,i,j)
• Inputs:
• Outputs:

## 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 :

• "symmetricMultiplication(LabeledModule,ZZ,ZZ)"

## For the programmer

The object symmetricMultiplication is .