next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
TensorComplexes :: symmetricMultiplication

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: Si(F)⊗Sj(F)→Si+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 :