Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| author | Escobar, Laura Fink, Alex Rajchgot, Jenna Woo, Alexander |
| author_facet | Escobar, Laura Fink, Alex Rajchgot, Jenna Woo, Alexander |
| contents | We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme-theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is $123$-avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley-Reisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_09589 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties Escobar, Laura Fink, Alex Rajchgot, Jenna Woo, Alexander Algebraic Geometry Commutative Algebra Combinatorics 05E40, 13P10, 14M15, 14N15 We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme-theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is $123$-avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley-Reisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams. |
| title | Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties |
| topic | Algebraic Geometry Commutative Algebra Combinatorics 05E40, 13P10, 14M15, 14N15 |
| url | https://arxiv.org/abs/2104.09589 |