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torusInvariants -- ring of invariants of torus action



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];
i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});

              2       4
o2 : Matrix ZZ  <-- ZZ
i3 : torusInvariants(T,R)

         2           2
o3 = QQ[y z, x*y*z, x z]

o3 : monomial subalgebra of R

See also

Ways to use torusInvariants :

For the programmer

The object torusInvariants is a method function with options.