# ambient(RingOfInvariants) -- the ambient polynomial ring where the group acted upon

## Synopsis

• Function: ambient
• Usage:
ambient S
• Inputs:
• S, an instance of the type RingOfInvariants, of the group action being returned
• Outputs:
• , where the group acted upon

## Description

This function is provided by the package InvariantRing.

This example shows how to recover the polynomial ring when a torus acted upon.

 i1 : R = QQ[x_1..x_4] o1 = R o1 : PolynomialRing i2 : T = diagonalAction(matrix {{0,1,-1,1},{1,0,-1,-1}}, R) * 2 o2 = R <- (QQ ) via | 0 1 -1 1 | | 1 0 -1 -1 | o2 : DiagonalAction i3 : S = R^T o3 = 2 QQ[x x x , x x x ] 1 2 3 1 3 4 o3 : RingOfInvariants i4 : ambient S o4 = R o4 : PolynomialRing