Description
Let T=(K^*)^r be the r-dimensional torus acting on the polynomial ring R=K[X_1,\ldots,X_n] diagonally. Such an action can be described as follows: there are integers a_{ij}, i=1,\ldots,r, j=1,\ldots,n, such that (\lambda_1,\ldots,\lambda_r)\in T acts by the substitution
X_j\mapsto \lambda_1^{a_{1j}}*\ldots*\lambda_r^{a_{rj}}X_j, j=1,\ldots,n.
The function takes the matrix (a_{ij}) as input and computes the ring of invariants R^T=\{f\in R: \lambda f=f for all \lambda \in T\}.
This method can be used with the options
allComputations and
grading.
i1 : R=QQ[x,y,z,w];
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i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});
2 4
o2 : Matrix ZZ <-- ZZ
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i3 : torusInvariants(T,R)
2 2
o3 = QQ[y z, x*y*z, x z]
o3 : monomial subalgebra of R
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