# torusInvariants -- ring of invariants of torus action

## Synopsis

• Usage:
torusInvariants(T,R)
• Inputs:
• , matrix (a_{ij}) of the action
• a ring, the ring on which the action takes place
• Outputs:

## 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

## Ways to use torusInvariants :

• "torusInvariants(Matrix,Ring)"

## For the programmer

The object torusInvariants is .