# Algebraic Group Actions on Polynomials

In the invariant theory package of Magma, a linear algebraic group $G$ is given by polynomials, say in variables $t_1,\ldots,t_m$, defined over some field $K$ that is representable in Magma, as the affine variety over the algebraic closure $\bar{K}$ of $K$ given by these polynomials. A $G$-module is given by a matrix $A \in K[t_1,\ldots,t_m]^{n \times n}$ such that a group element $(\eta_1,\ldots,\eta_m) \in G$ acts on $\bar{K}^n$ by the matrix obtained by substituting $(\eta_1,\ldots,\eta_m)$ into the polynomials occurring in the matrix $A$.

$G$ then also acts on the ring of polynomials on $\bar{K}^n$ by

$$
\sigma(f) = f \circ \sigma^{-1}
$$

for $\sigma \in G$ and $f \in \bar{K}[x_1,\ldots,x_n]$. Since the algorithms in Magma do not work with the algebraic closure, single group elements are never dealt with. In fact, all relevant algorithms ony involve field elements of $K$, the field of definition.