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# ToricVectorBundle ** ToricVectorBundle -- the tensor product of two toric vector bundles

- Operator: **
- Usage:
    `E = E1 ** E2`
- Inputs:
  - E1, an instance of the type ToricVectorBundle
  - E2, an instance of the type ToricVectorBundle
- Outputs:
  - E, an instance of the type ToricVectorBundle

## Description

If $E_1$ and $E_2$ are defined over the same fan and in the same description, then tensor computes the tensor product of the two vector bundles in this description

    i1 : E1 = toricVectorBundle(2,hirzebruchFan 3)

    o1 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 2

    o1 : ToricVectorBundleKlyachko

    i2 : E2 = tangentBundle hirzebruchFan 3

    o2 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 2

    o2 : ToricVectorBundleKlyachko

    i3 : E = E1 ** E2

    o3 = {dimension of the variety => 2 }
          number of affine charts => 4
          number of rays => 4
          rank of the vector bundle => 4

    o3 : ToricVectorBundleKlyachko

    i4 : details E

    o4 = HashTable{| -1 | => (| -1 1/3 0  0   |, | -1 0 -1 0 |)}
                   | 3  |     | 3  0   0  0   |
                              | 0  0   -1 1/3 |
                              | 0  0   3  0   |
                   | 0  | => (| 0  1 0  0 |, | -1 0 -1 0 |)
                   | -1 |     | -1 0 0  0 |
                              | 0  0 0  1 |
                              | 0  0 -1 0 |
                   | 0 | => (| 0 1 0 0 |, | -1 0 -1 0 |)
                   | 1 |     | 1 0 0 0 |
                             | 0 0 0 1 |
                             | 0 0 1 0 |
                   | 1 | => (| 1 0 0 0 |, | -1 0 -1 0 |)
                   | 0 |     | 0 1 0 0 |
                             | 0 0 1 0 |
                             | 0 0 0 1 |

    o4 : HashTable

    i5 : E1 = toricVectorBundle(2,hirzebruchFan 3,"Type" => "Kaneyama")

    o5 = {dimension of the variety => 2 }
          number of affine charts => 4
          rank of the vector bundle => 2

    o5 : ToricVectorBundleKaneyama

    i6 : E2 = tangentBundle(hirzebruchFan 3,"Type" => "Kaneyama")

    o6 = {dimension of the variety => 2 }
          number of affine charts => 4
          rank of the vector bundle => 2

    o6 : ToricVectorBundleKaneyama

    i7 : E = E1 ** E2

    o7 = {dimension of the variety => 2 }
          number of affine charts => 4
          rank of the vector bundle => 4

    o7 : ToricVectorBundleKaneyama

    i8 : details E

    o8 = (HashTable{0 => (| 0 -1 |, | 1 1 -3 -3 |) }, HashTable{(0, 1) => |
                          | 1 3  |  | 0 0 -1 -1 |                         |
                    1 => (| 0  -1 |, | 1 1 3 3 |)                         |
                          | -1 3  |  | 0 0 1 1 |                          |
                    2 => (| 1 0 |, | -1 -1 0  0  |)             (0, 2) => |
                          | 0 1 |  | 0  0  -1 -1 |                        |
                    3 => (| 1 0  |, | -1 -1 0 0 |)                        |
                          | 0 -1 |  | 0  0  1 1 |                         |
                                                                (1, 3) => |
                                                                          |
                                                                          |
                                                                          |
                                                                (2, 3) => |
                                                                          |
                                                                          |
                                                                          |
         ------------------------------------------------------------------------
         1 0 0  0  |})
         0 1 0  0  |
         0 0 -1 0  |
         0 0 0  -1 |
         -1 0  0 0 |
         0  -1 0 0 |
         3  0  1 0 |
         0  3  0 1 |
         -1 0  0 0 |
         0  -1 0 0 |
         -3 0  1 0 |
         0  -3 0 1 |
         1 0 0  0  |
         0 1 0  0  |
         0 0 -1 0  |
         0 0 0  -1 |

    o8 : Sequence

## See also

- tensor(ToricVectorBundle,ToricVectorBundle) -- the tensor product of two toric vector bundles
- ToricVectorBundle ++ ToricVectorBundle -- the direct sum of two toric vector bundles
- exteriorPower(ZZ,ToricVectorBundle) -- the 'l'-th exterior power of a toric vector bundle
- symmetricPower(ZZ,ToricVectorBundle) -- the 'l'-th symmetric power of a toric vector bundle

## Ways to use this method:

- ToricVectorBundle ** ToricVectorBundle -- the tensor product of two toric vector bundles

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:3492:0.