Lifting Gröbner Bases from the Exterior Algebra

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ⟨X ₁,…, X _n ⟩ which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ⟨X ₁,…, X _n ⟩, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.

Key Words:

• Exterior algebra
• Free associative algebra
• Gröbner basis
• Generic initial ideal

Acknowledgments

The first author was partially supported by a post-doc fellowship from the European Commission within the European TMR Network “Harmonic Analysis” 1998–2001 (Contract ERBFMRX-CT 97-0159) and partially by the JSPS.

The second author was supported by a grant from Svenska Institutet and by grant n. 231801F from Centre International des Etudiants et Stagiaires.

We thank the referee for many valuable comments.

Notes

^aIf 𝔞 contains the anti-commutator ideal, then in (g(𝔞)) is finitely generated, not only for a generic g ∈ GL(V), but for all g ∈ GL(V).