Gröbner bases for bipartite determinantal ideals
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911928964612096 |
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| author | Illian, Josua Li, Li |
| author_facet | Illian, Josua Li, Li |
| contents | Nakajima's graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals.
We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia--Guccione--Guccione.
As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01724 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gröbner bases for bipartite determinantal ideals Illian, Josua Li, Li Commutative Algebra Algebraic Geometry Primary 14M12, Secondary 13P10, 13C40, 05E40 Nakajima's graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals. We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia--Guccione--Guccione. As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics. |
| title | Gröbner bases for bipartite determinantal ideals |
| topic | Commutative Algebra Algebraic Geometry Primary 14M12, Secondary 13P10, 13C40, 05E40 |
| url | https://arxiv.org/abs/2305.01724 |