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# flip -- isomorphism map of commutativity of tensor product

- Usage:
    `flip(F, G)`
- Inputs:
  - F, a module,
  - G, a module,
- Outputs:
  - a matrix, the matrix representing the natural isomorphism $F \otimes G \to G \otimes F$

## Description

    i1 : R = QQ[x,y];

    i2 : F = R^{1,2,3}

          3
    o2 = R

    o2 : R-module, free, degrees {-1, -2, -3}

    i3 : G = R^{10,20,30}

          3
    o3 = R

    o3 : R-module, free, degrees {-10, -20, -30}

    i4 : f = flip(F,G)

    o4 = {-11} | 1 0 0 0 0 0 0 0 0 |
         {-12} | 0 0 0 1 0 0 0 0 0 |
         {-13} | 0 0 0 0 0 0 1 0 0 |
         {-21} | 0 1 0 0 0 0 0 0 0 |
         {-22} | 0 0 0 0 1 0 0 0 0 |
         {-23} | 0 0 0 0 0 0 0 1 0 |
         {-31} | 0 0 1 0 0 0 0 0 0 |
         {-32} | 0 0 0 0 0 1 0 0 0 |
         {-33} | 0 0 0 0 0 0 0 0 1 |

                 9      9
    o4 : Matrix R  <-- R

    i5 : isHomogeneous f

    o5 = true

    i6 : target f

          9
    o6 = R

    o6 : R-module, free, degrees {-11, -12, -13, -21, -22, -23, -31, -32, -33}

    i7 : source f

          9
    o7 = R

    o7 : R-module, free, degrees {-11, -21, -31, -12, -22, -32, -13, -23, -33}

    i8 : target f === G**F

    o8 = true

    i9 : source f === F**G

    o9 = true

    i10 : u = x * F_0

    o10 = | x |
          | 0 |
          | 0 |

           3
    o10 : R

    i11 : v = y * G_1

    o11 = | 0 |
          | y |
          | 0 |

           3
    o11 : R

    i12 : u ** v

    o12 = |  0 |
          | xy |
          |  0 |
          |  0 |
          |  0 |
          |  0 |
          |  0 |
          |  0 |
          |  0 |

           9
    o12 : R

    i13 : v ** u

    o13 = |  0 |
          |  0 |
          |  0 |
          | xy |
          |  0 |
          |  0 |
          |  0 |
          |  0 |
          |  0 |

           9
    o13 : R

    i14 : f * (u ** v)

    o14 = |  0 |
          |  0 |
          |  0 |
          | xy |
          |  0 |
          |  0 |
          |  0 |
          |  0 |
          |  0 |

           9
    o14 : R

    i15 : f * (u ** v) === v ** u

    o15 = true

## Ways to use flip:

- `flip(Module,Module)`

## For the programmer

The object flip is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/tensor-doc.m2:429:0.