# degreesRing(DiagonalAction) -- of a diagonal action

## Synopsis

• Function: degreesRing
• Usage:
degreesRing D
• Inputs:
• Outputs:
• a ring, where the equivariant Hilbert series of the diagonal group action lives in

## Description

This function is provided by the package InvariantRing.

Use this function to get the ring where the equivariant Hilbert series of the diagonal group action lives in.

The following example defines an action of the product of a two-dimensional torus and two cyclic group of order 3 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 : W1 = matrix{{1,0,1,0},{0,1,1,0}} o3 = | 1 0 1 0 | | 0 1 1 0 | 2 4 o3 : Matrix ZZ <--- ZZ i4 : T = diagonalAction(W,W1,{3,3},R) * 2 o4 = R <- (QQ ) x ZZ/3 x ZZ/3 via (| 0 1 -1 1 |, | 1 0 1 0 |) | 1 0 -1 -1 | | 0 1 1 0 | o4 : DiagonalAction i5 : degreesRing T o5 = ZZ[z ..z ][T] 0 3 o5 : PolynomialRing