invariantRing -- the ring of invariants of a group action

Synopsis

• Usage:

  invariantRing G, R^G

• Inputs:
  • G, an instance of the type GroupAction
  • R, a polynomial ring, on which the group acts
• Optional inputs:
  • Strategy (missing documentation) => ..., default value UseNormaliz, the strategy used to compute the invariant ring
  • DegreeBound => ..., default value infinity, degree bound for invariants of finite groups
  • UseCoefficientRing => ..., default value false, option to compute invariants over the given coefficient ring
  • UseLinearAlgebra => ..., default value false, strategy for computing invariants of finite groups
• Outputs:
  • an instance of the type RingOfInvariants, the ring of invariants of the given group action

Description

The following example defines an action of a two-dimensional torus on a four-dimensional vector space with a basis of weight vectors whose weights are the columns of the input matrix.

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 : S = invariantRing T

o4 =             2
     QQ[x x x , x x x ]
         1 2 3   1 3 4

o4 : RingOfInvariants

The algebra generators for the ring of invariants are computed upon initialization by the method invariants.

Alternatively, we can use the following shortcut to construct a ring of invariants.

i5 : S = R^T

o5 =             2
     QQ[x x x , x x x ]
         1 2 3   1 3 4

o5 : RingOfInvariants

Caveat