Commutative algebra and algebraic varieties
This report surveys the issue of finding the rank varieties that arise from the representation of the permutation groups Sn over the finite field Fp. The underlying module that we are working with is the simple module D(p − 1) for p a prime number. Here we explain how the rank varieties are computed...
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| Format: | Final Year Project (FYP) |
| Language: | English |
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Nanyang Technological University
2021
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| Online Access: | https://hdl.handle.net/10356/148497 |
| _version_ | 1824456656411426816 |
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| author | Tan, Zhi Hong |
| author2 | Lim Kay Jin |
| author_facet | Lim Kay Jin Tan, Zhi Hong |
| author_sort | Tan, Zhi Hong |
| collection | NTU |
| description | This report surveys the issue of finding the rank varieties that arise from the representation of the permutation groups Sn over the finite field Fp. The underlying module that we are working with is the simple module D(p − 1) for p a prime number. Here we explain how the rank varieties are computed, followed by analyzing its reducibility. Letting n = pk, for small cases p = 2, k = 2, 3, 4 and p = 3, k = 2, 3, we found that the rank varieties obtained are homogeneous multivariate polynomials which take on the form of ∑(Xi1, Xi2, ···, Xik−1)^{p−1}, where the index are all possible strictly increasing sequences satisfying 1 ≤ i1 < i2 < ··· < ik−1 ≤ k. We also established the irreducibility of the rank variety for the case p = 3, k = 3. Lastly, we explored the Hilbert Series of the quotient of the polynomial ring by the rank varieties. It turns out that the Hilbert Series depends on the degree of the homogeneous polynomial associated to the rank variety. |
| first_indexed | 2025-02-19T03:57:34Z |
| format | Final Year Project (FYP) |
| id | ntu-10356/148497 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2025-02-19T03:57:34Z |
| publishDate | 2021 |
| publisher | Nanyang Technological University |
| record_format | dspace |
| spelling | ntu-10356/1484972023-02-28T23:19:39Z Commutative algebra and algebraic varieties Tan, Zhi Hong Lim Kay Jin Tan Geok Choo School of Physical and Mathematical Sciences LimKJ@ntu.edu.sg, GCTan@ntu.edu.sg Science::Mathematics::Algebra This report surveys the issue of finding the rank varieties that arise from the representation of the permutation groups Sn over the finite field Fp. The underlying module that we are working with is the simple module D(p − 1) for p a prime number. Here we explain how the rank varieties are computed, followed by analyzing its reducibility. Letting n = pk, for small cases p = 2, k = 2, 3, 4 and p = 3, k = 2, 3, we found that the rank varieties obtained are homogeneous multivariate polynomials which take on the form of ∑(Xi1, Xi2, ···, Xik−1)^{p−1}, where the index are all possible strictly increasing sequences satisfying 1 ≤ i1 < i2 < ··· < ik−1 ≤ k. We also established the irreducibility of the rank variety for the case p = 3, k = 3. Lastly, we explored the Hilbert Series of the quotient of the polynomial ring by the rank varieties. It turns out that the Hilbert Series depends on the degree of the homogeneous polynomial associated to the rank variety. Bachelor of Science in Mathematical Sciences 2021-04-28T02:52:27Z 2021-04-28T02:52:27Z 2021 Final Year Project (FYP) Tan, Z. H. (2021). Commutative algebra and algebraic varieties. Final Year Project (FYP), Nanyang Technological University, Singapore. https://hdl.handle.net/10356/148497 https://hdl.handle.net/10356/148497 en application/pdf Nanyang Technological University |
| spellingShingle | Science::Mathematics::Algebra Tan, Zhi Hong Commutative algebra and algebraic varieties |
| title | Commutative algebra and algebraic varieties |
| title_full | Commutative algebra and algebraic varieties |
| title_fullStr | Commutative algebra and algebraic varieties |
| title_full_unstemmed | Commutative algebra and algebraic varieties |
| title_short | Commutative algebra and algebraic varieties |
| title_sort | commutative algebra and algebraic varieties |
| topic | Science::Mathematics::Algebra |
| url | https://hdl.handle.net/10356/148497 |
| work_keys_str_mv | AT tanzhihong commutativealgebraandalgebraicvarieties |