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 |
The object invariantRing is a method function with options.