# invariantRing -- the ring of invariants of a group action

## Synopsis

• Usage:
invariantRing G, R^G
• Inputs:
• G, an instance of the type GroupAction
• R, , 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

This function is provided by the package InvariantRing.

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

By default, the invariants of a diagonal group action are computed over an infinite extension of the ground field specified by the user over which the action is defined. Provided the action is defined, it is possible to compute invariants literally over the specified ground field in prime characteristic using the option UseCoefficientRing.

## Ways to use invariantRing :

• "invariantRing(GroupAction)"

## For the programmer

The object invariantRing is .