Bibliographic Information: Burciu, S. (2024). Frobenius-Perron dimensions of conjugacy classes in fusion categories [Preprint]. arXiv:2410.05709v1.
Research Objective: This research paper aims to explore the relationship between the arithmetic properties of conjugacy class sizes in weakly integral fusion categories and the structure of the categories themselves, drawing inspiration from similar investigations in finite group theory.
Methodology: The author utilizes the framework of pivotal fusion categories, employing tools such as Harada's identity on the product of conjugacy class sums, the canonical pairing between class functions and central elements, and the concept of support of a fusion subcategory. The study leverages previous results on the structure constants of premodular categories and properties of nilpotent fusion categories.
Key Findings: The paper establishes several key findings:
Theorem 1.1: In a weakly integral braided fusion category, a specific expression involving the Frobenius-Perron dimensions of conjugacy classes and the maximal pointed fusion subcategory is proven to be an integer.
Theorem 1.2: If a prime p divides the Frobenius-Perron dimension of a weakly integral braided fusion category but not the dimension of any conjugacy class, then a power of p divides the dimension of the maximal pointed fusion subcategory.
Corollary 1.3: For weakly-integral modular tensor categories, if a prime p divides the category's dimension but not the dimension of any simple object, then a power of p divides the order of the universal grading group.
Theorem 1.4: In a braided nilpotent fusion category, the Frobenius-Perron dimension of any conjugacy class divides a specific ratio involving the dimensions of the category and its maximal pointed fusion subcategory.
Main Conclusions: The results presented in this paper provide new insights into the structure of weakly integral fusion categories, particularly by examining the Frobenius-Perron dimensions of their conjugacy classes. The findings highlight intriguing connections between the arithmetic properties of these dimensions and the categorical structure, mirroring similar phenomena observed in finite group theory.
Significance: This research contributes significantly to the field of fusion category theory by extending classical results from finite group representation theory to a more general categorical setting. The established divisibility properties and relationships between dimensions offer valuable tools for further exploration and classification of fusion categories.
Limitations and Future Research: The study primarily focuses on weakly integral fusion categories. Exploring similar properties in more general fusion categories could be a promising avenue for future research. Additionally, investigating whether stronger parallels can be drawn with the Ito-Michler theorem from finite group theory, potentially without relying on the classification of finite simple groups, presents an exciting challenge.
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