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):binB. Perhaps we also wish to form the set
IG(B):=V:VtextisanirreduciblesummandofW,textforsomeWinG(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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