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Summary
The paper explores the concept of "algebraic self-similarity" on a 4D cubic lattice, demonstrating that a 2x2x2x2 block of matrices A can be decomposed into four direct "block spin" summands corresponding to the same matrix A.
Highlights
- The 4D cubic lattice has a "spin" at each edge and a copy of a 4x4 matrix A at each vertex.
- The block B made of 16 copies of A reveals four direct "block spin" summands corresponding to A.
- The summands can be written in an elegant way using the Frobenius transform of A's entries.
- When A's entries are zeros and ones, two more "block spins" split out with different A's.
- The phenomena were discovered using computer algebra packages GAP and Singular.
- The results may lead to the calculation of interesting combinatorial correlations.
- The algebraic structure of the block B is more complicated than expected.
Key Insights
- The concept of "algebraic self-similarity" is extended to a 4D cubic lattice, demonstrating the decomposition of a 2x2x2x2 block of matrices A into four direct "block spin" summands corresponding to A.
- The block B made of 16 copies of A has a more complicated algebraic structure than expected, with the Frobenius transform playing a crucial role in the decomposition.
- When A's entries are restricted to zeros and ones, additional "block spins" emerge, corresponding to different matrices A, which may lead to new combinatorial correlations.
- The use of computer algebra packages GAP and Singular was instrumental in discovering these phenomena, highlighting the importance of computational tools in algebraic research.
- The results have potential applications in statistical physics, particularly in the study of phase transitions and critical phenomena.
- The algebraic structure of the block B is a direct sum of four linear spaces, each corresponding to a "block spin" summand, which may lead to new insights into the behavior of the 4D cubic lattice.
- The study of the 4D cubic lattice's algebraic structure may also shed light on the properties of other lattice models, such as the 3D Ising model.
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Citation
Korepanov, I. G. (2024). Self-similarity on 4d cubic lattice (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.20140