Polynomial invariants of a rational algebraic-group action

Let $k$ be a field. An affine algebraic group $G$ over $k$ is an affine variety with algebraic multiplication, identity, and inverse maps satisfying the group axioms. Its coordinate algebra $k[G]$ has comultiplication $\mathrm{\Delta }$ and counit $\epsilon$, the pullbacks of multiplication and the identity map. A finite-dimensional rational left $G$-module is a finite-dimensional $k$-vector space $V$ with a $k$-linear coaction ${\lambda }_V:V\to k[G]{\otimes }_{k}V$ satisfying $(\mathrm{\Delta }\otimes {\mathrm{id}}_V){\lambda }_V=({\mathrm{id}}_{k[G]}\otimes {\lambda }_V){\lambda }_V$ and $(\epsilon \otimes {\mathrm{id}}_V){\lambda }_V={\mathrm{id}}_V$; equivalently it is a morphism of varieties $G\times V\to V$ defining an algebraic left action. With this left-comodule convention the left action is recovered by $g\cdot v=({\mathrm{ev}}_{g^{-1}}\otimes {\mathrm{id}}_V){\lambda }_V(v)$; equivalently ${\lambda }_V(v)$ evaluated at $g$ is $g^{-1}\cdot v$. The inverse is necessary because evaluation of the displayed left coaction directly at $g$ gives a right action. Its affine coordinate ring is $k[V]={\mathrm{Sym}}_k(V^{\ast })$. For every integer $t\ge 0$, the diagonal action on $V^t$ induces a coaction ${\lambda }_{k[V^t]}:k[V^t]\to k[G]{\otimes }_{k}k[V^t]$ by pullback of the action morphism. The invariant subring is $k[V^t{]}^G=\{f\in k[V^t]:{\lambda }_{k[V^t]}(f)=1\otimes f\}\mathrm{.}$ This is the algebraic-group coaction convention; it does not identify invariants of the finite group of $k$-points $G(k)$ with $k[V^t{]}^G$.