Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.16364 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915033217236992 |
|---|---|
| author | Koley, Mitra Kotal, Nirmal Veer, Dharm |
| author_facet | Koley, Mitra Kotal, Nirmal Veer, Dharm |
| contents | In this article, we study two fundamental questions on polyomino ideals which are radicality and primality. In order to study the question of radicality, we initiate the study of Knutson ideals among polyominoes. Knutson ideals were introduced by Conca and Varbaro after the work of Knutson on compatibly split ideals. Knutson ideals are known to have nice properties, for example, they are well behaved with Gröbner bases, and it has square-free initial ideals; hence they are radical.
We show that polyomino ideals associated with closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson. We also show that polyomino ideals associated with a class of thin polyominoes are Knutson; hence they are radical. In fact, we show that these polyomino ideals are prime and the reduced Gröbner basis is computed. Furthermore, we prove that under a certain condition, if a parallelogram polyomino is extracted from another parallelogram polyomino, the resulting collection of cells is Knutson. We also compute their Gröbner basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polyominoes and Knutson ideals Koley, Mitra Kotal, Nirmal Veer, Dharm Commutative Algebra 05E40, 05B50, 13P10 In this article, we study two fundamental questions on polyomino ideals which are radicality and primality. In order to study the question of radicality, we initiate the study of Knutson ideals among polyominoes. Knutson ideals were introduced by Conca and Varbaro after the work of Knutson on compatibly split ideals. Knutson ideals are known to have nice properties, for example, they are well behaved with Gröbner bases, and it has square-free initial ideals; hence they are radical. We show that polyomino ideals associated with closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson. We also show that polyomino ideals associated with a class of thin polyominoes are Knutson; hence they are radical. In fact, we show that these polyomino ideals are prime and the reduced Gröbner basis is computed. Furthermore, we prove that under a certain condition, if a parallelogram polyomino is extracted from another parallelogram polyomino, the resulting collection of cells is Knutson. We also compute their Gröbner basis. |
| title | Polyominoes and Knutson ideals |
| topic | Commutative Algebra 05E40, 05B50, 13P10 |
| url | https://arxiv.org/abs/2411.16364 |