# weights -- of a diagonal action

## Synopsis

• Usage:
weights D
• Inputs:
• Outputs:
• , the weight matrix of the group action

## Description

This function is provided by the package InvariantRing.

Use this function to recover the weight matrix of a diagonal action on a polynomial ring. For a diagonal action on a polynomial ring $k[x_1, \dots, x_n]$ , the $j$ -th column of the weight matrix is the weight of the variable $x_j$ .

The following example defines an action of a two-dimensional torus on a polynomial ring in four variables.

 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 : weights T o4 = (| 0 1 -1 1 |, 0) | 1 0 -1 -1 | o4 : Sequence

Here is an example of a product of two cyclic groups of order 3 acting on a polynomial ring in 3 variables.

 i5 : R = QQ[x_1..x_3] o5 = R o5 : PolynomialRing i6 : d = {3,3} o6 = {3, 3} o6 : List i7 : W = matrix{{1,0,1},{0,1,1}} o7 = | 1 0 1 | | 0 1 1 | 2 3 o7 : Matrix ZZ <--- ZZ i8 : A = diagonalAction(W, d, R) o8 = R <- ZZ/3 x ZZ/3 via | 1 0 1 | | 0 1 1 | o8 : DiagonalAction i9 : weights A o9 = (0, | 1 0 1 |) | 0 1 1 | o9 : Sequence