torusInvariants -- ring of invariants of torus action

Synopsis

• Usage:

  torusInvariants(T,R)

• Inputs:
  • a matrix, matrix (a_{ij}) of the action
  • a ring, the ring on which the action takes place
• Outputs:
  • an instance of the type MonomialSubalgebra, the ring of invariants R^T

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];

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

• finiteDiagInvariants -- ring of invariants of a finite group action
• diagInvariants -- ring of invariants of a diagonalizable group action