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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 $B$?

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    $\begingroup$ Note that no change to the standard theory is necessary if $G$ is a torus -- now we're just talking multigraded Hilbert series, rather than usual. $\endgroup$ – Allen Knutson Jan 17 '10 at 13:15
  • $\begingroup$ You've probably already found this, but this paper sets up a theory of equivariant Gröbner bases, though I think not quite of the type you seek. $\endgroup$ – Clark Barwick Jan 18 '10 at 15:09
  • $\begingroup$ I didn't know about this, and it looks quite interesting even if it's not exactly what I'm asking for. Thank you! $\endgroup$ – Jonah Blasiak Jan 19 '10 at 0:22
  • $\begingroup$ Any luck? I'd also like to hear about this! $\endgroup$ – Clark Barwick Feb 4 '10 at 15:26
  • $\begingroup$ Two comments: (1) To get braces to appear, you have to use two backslashes: \\{, provided Markdown doesn't edit this comment. This is because Markdown converts backslash-brace to brace, so to get backslash-brace, you must use backslash-backslash-brace. (2) Clark, you should post your link as an answer, and if it's what Jonah was looking for, you should accept it, and otherwise maybe comment on it? $\endgroup$ – Theo Johnson-Freyd Feb 6 '10 at 3:39
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Have you seen the old paper Symbolic solution of polynomial equation systems with symmetry by Karin Gatermann? Or Zeros of equivariant vector fields: Algorithms for an invariant approach by Patrick Worfolk?

There is also the more recent work Equivariant Groebner bases and the Gaussian two-factor model by Brouwer and Draisma, as well as Solving Systems of Polynomial Equations with Symmetries Using SAGBI-Gröbner Bases by Faugere and Rahmany?

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You might want to look at the paper Groebner bases of ideals invariant under endomorphisms by Drensky and La Scala.

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