# actionMatrix -- matrix of a linearly reductive action

## Synopsis

• Usage:
actionMatrix L
• Inputs:
• Outputs:
• , of a group action on a ring

## Description

This function is provided by the package InvariantRing.

Suppose the action L consists of the linearly reductive group with coordinate ring S/I (where S is a polynomial ring) acting on a (quotient of) a polynomial ring R via the action matrix M. This function returns the action matrix M.

 i1 : S = QQ[z] o1 = S o1 : PolynomialRing i2 : I = ideal(z^2 - 1) 2 o2 = ideal(z - 1) o2 : Ideal of S i3 : M = matrix{{(z+1)/2, (1-z)/2},{(1-z)/2, (z+1)/2}} o3 = | 1/2z+1/2 -1/2z+1/2 | | -1/2z+1/2 1/2z+1/2 | 2 2 o3 : Matrix S <--- S i4 : R = QQ[x,y] o4 = R o4 : PolynomialRing i5 : L = linearlyReductiveAction(I, M, R) 2 o5 = R <- S/ideal(z - 1) via | 1/2z+1/2 -1/2z+1/2 | | -1/2z+1/2 1/2z+1/2 | o5 : LinearlyReductiveAction i6 : actionMatrix L o6 = | 1/2z+1/2 -1/2z+1/2 | | -1/2z+1/2 1/2z+1/2 | 2 2 o6 : Matrix S <--- S

## Ways to use actionMatrix :

• "actionMatrix(LinearlyReductiveAction)"

## For the programmer

The object actionMatrix is .