Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912126941003776 |
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| author | Gesmundo, Fulvio Hang Huang Schenck, Hal Weyman, Jerzy |
| author_facet | Gesmundo, Fulvio Hang Huang Schenck, Hal Weyman, Jerzy |
| contents | We describe the minimal free resolution of the ideal of $2 \times 2$ subpermanents of a $2 \times n$ generic matrix $M$. In contrast to the case of $2 \times 2$ determinants, the $2 \times 2$ permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of $2 \times 2$ permanents of a generic matrix with our previous work connecting the initial ideal of $2 \times 2$ permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_12247 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix Gesmundo, Fulvio Hang Huang Schenck, Hal Weyman, Jerzy Commutative Algebra Algebraic Geometry 13D02, 13F55, 13C40, 68Q15 We describe the minimal free resolution of the ideal of $2 \times 2$ subpermanents of a $2 \times n$ generic matrix $M$. In contrast to the case of $2 \times 2$ determinants, the $2 \times 2$ permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of $2 \times 2$ permanents of a generic matrix with our previous work connecting the initial ideal of $2 \times 2$ permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence. |
| title | Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix |
| topic | Commutative Algebra Algebraic Geometry 13D02, 13F55, 13C40, 68Q15 |
| url | https://arxiv.org/abs/2312.12247 |