Commutative algebra and algebraic varieties

• Main Author: Tan, Zhi Hong
• Other Authors: Lim Kay Jin
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2021
• Subjects: Science::Mathematics::Algebra
• Online Access: https://hdl.handle.net/10356/148497
• Institution: Nanyang Technological University

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.