weights -- of a diagonal action

Synopsis

• Usage:

  weights D

• Inputs:
  • D, an instance of the type DiagonalAction
• Outputs:
  • a matrix, the weight matrix of the group action

Description

Use this function to recover the weight matrix of a diagonal action on a polynomial ring. For a diagonal action on a polynomial ring $k[x_1, \dots, x_n]$ , the $j$ -th column of the weight matrix is the weight of the variable $x_j$ .

The following example defines an action of a two-dimensional torus on a polynomial ring in four variables.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : W = matrix{{0,1,-1,1},{1,0,-1,-1}}

o2 = | 0 1 -1 1  |
     | 1 0 -1 -1 |

              2       4
o2 : Matrix ZZ  <-- ZZ

i3 : T = diagonalAction(W, R)

             * 2
o3 = R <- (QQ )  via 

     | 0 1 -1 1  |
     | 1 0 -1 -1 |

o3 : DiagonalAction

i4 : weights T

o4 = (| 0 1 -1 1  |, 0)
      | 1 0 -1 -1 |

o4 : Sequence

Here is an example of a product of two cyclic groups of order 3 acting on a polynomial ring in 3 variables.

i5 : R = QQ[x_1..x_3]

o5 = R

o5 : PolynomialRing

i6 : d = {3,3}

o6 = {3, 3}

o6 : List

i7 : W = matrix{{1,0,1},{0,1,1}}

o7 = | 1 0 1 |
     | 0 1 1 |

              2       3
o7 : Matrix ZZ  <-- ZZ

i8 : A = diagonalAction(W, d, R)

o8 = R <- ZZ/3 x ZZ/3 via 

     | 1 0 1 |
     | 0 1 1 |

o8 : DiagonalAction

i9 : weights A

o9 = (0, | 1 0 1 |)
         | 0 1 1 |

o9 : Sequence

See also

• DiagonalAction -- the class of all diagonal actions
• diagonalAction -- diagonal group action via weights