adjoint'(Matrix,Module,Module) -- an adjoint map

Synopsis

• Function: adjoint'

• Usage:

  adjoint'(f,G,H)

• Inputs:
  • f, a matrix, a homomorphism F --> Hom(G,H) between modules
  • G, a module, a free module
  • H, a module, a free module
• Outputs:
  • a matrix, the adjoint homomorphism F ** G --> H

Description

Recall that ** refers to the tensor product of modules. If f is homogeneous, then the resulting matrix will be homogeneous.

i1 : R = QQ[x_1 .. x_12];

i2 : f = genericMatrix(R,6,2)

o2 = | x_1 x_7  |
     | x_2 x_8  |
     | x_3 x_9  |
     | x_4 x_10 |
     | x_5 x_11 |
     | x_6 x_12 |

             6       2
o2 : Matrix R  <--- R

i3 : g = adjoint'(f,R^2,R^3)

o3 = | x_1 x_4 x_7 x_10 |
     | x_2 x_5 x_8 x_11 |
     | x_3 x_6 x_9 x_12 |

             3       4
o3 : Matrix R  <--- R

i4 : isHomogeneous g

o4 = true

See also

• adjoint -- an adjoint map
• flip -- matrix of commutativity of tensor product
• reshape -- reshape a matrix
• Module ** Module -- tensor product
• dual -- dual module or map