At one point my advisor, Mark Haiman, mentioned that it would be nice if there was a way to compute Groebner bases that takes into account a group action.

Does anyone know of any work done along these lines?

For example, suppose a general linear group $G$ acts on a polynomial ring $R$ and we have an ideal $I$ invariant under the group action. Suppose we have a Groebner basis $B$ of $I$. Then we can form the set $G(B) := \{ G(b) : b \in B \}$. Perhaps we also wish to form the set

$$IG(B) := \{ V : V \text{ is an irreducible summand of } W, \text{ for some }W \in G(B) \}$$
(note that $G(b)$ cyclic implies it has a multiplicity-free decomposition into irreducibles).

Can we find a condition on a set of $G$-modules (resp. $G$-irreducibles), analogous to Buchberger's S-pair criterion, that guarantees that this set is of the form $G(B)$ (resp. $IG(B)$) for some Groebner basis $B$?

Can the character of $R /I$ be determined from the set $IG(B)$ in a similar way to how the Hilbert series of $R /I$ can be determined from $B$?