numgens(DiagonalAction) -- number of generators of the finite part of a diagonal group

Synopsis

• Function: numgens

• Usage:

  numgens D

• Inputs:
  • D, an instance of the type DiagonalAction, the action of a diagonal group
• Outputs:
  • an integer, the number of generators of the group

Description

Writing the diagonal group acting on the polynomial ring $k[x_1,\dots,x_n]$ as $(k^*)^r \times \mathbb{Z}/d_1 \times \cdots \times \mathbb{Z}/d_g$, this function returns g.

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

i1 : R = QQ[x_1..x_3]

o1 = R

o1 : PolynomialRing

i2 : d = {3,3}

o2 = {3, 3}

o2 : List

i3 : W = matrix{{1,0,1},{0,1,1}}

o3 = | 1 0 1 |
     | 0 1 1 |

              2       3
o3 : Matrix ZZ  <-- ZZ

i4 : A = diagonalAction(W, d, R)

o4 = R <- ZZ/3 x ZZ/3 via 

     | 1 0 1 |
     | 0 1 1 |

o4 : DiagonalAction

i5 : numgens A

o5 = 2