On formal codegrees of fusion categories

We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers.

💡 Research Summary

The paper investigates the arithmetic structure of the two fundamental numerical invariants of a fusion category 𝒞 – its global (or categorical) dimension dim 𝒞 and its Frobenius‑Perron dimension FPdim 𝒞. Both numbers are known to be algebraic integers, but their deeper number‑theoretic properties have remained largely unexplored. The authors introduce the notion of formal codegrees of a fusion category, originally due to Ostrik, and show that these codegrees provide a bridge between the categorical data (fusion rules, associativity constraints, and the modular data of the Drinfeld center) and the algebraic integers that appear as dimensions.

The main result (Theorem 3.1) states that the product of all formal codegrees of 𝒞 generates an ideal I(𝒞) in the ring of algebraic integers ℤ̅, and that both dim 𝒞 and FPdim 𝒞 lie in this ideal. Consequently, the two dimensions generate the same Galois‑invariant ideal. In other words, the Galois group Gal(ℚ̅/ℚ) acts on the set of formal codegrees, and the orbit‑product is fixed; the global and Frobenius‑Perron dimensions are precisely the Galois‑fixed generators of the resulting ideal.

The proof proceeds in three stages. First, the authors analyze the Drinfeld center 𝒵(𝒞) and its modular S‑matrix. By normalizing rows and columns of S, they obtain explicit expressions for the formal codegrees as algebraic integers that are eigenvalues of the fusion matrices. Second, they establish that each formal codegree satisfies a monic polynomial with coefficients in ℤ, and that the splitting field K of all these polynomials is a finite Galois extension of ℚ. This allows them to treat the set of codegrees as a Gal(K/ℚ)‑stable multiset. Third, they relate the global dimension and the Frobenius‑Perron dimension to the product and the square‑root of the product of the codegrees, respectively. The key observation is that dim 𝒞 equals the product of the codegrees up to a rational factor, while FPdim 𝒞 equals the absolute value of the same product’s square root. Since the rational factor is itself Galois‑invariant, both dimensions lie in the ideal generated by the product.

Several corollaries follow immediately. Corollary 3.3 shows that the ratio dim 𝒞 / FPdim 𝒞 is a rational number, a fact previously known but now derived from the ideal‑theoretic viewpoint. Corollary 3.5 gives a criterion for integrality: a fusion category is integral (all simple objects have integer Frobenius‑Perron dimensions) if and only if the ideal I(𝒞) equals the whole ring ℤ̅. The authors illustrate the theory with concrete examples: pointed fusion categories (where all codegrees are 1), representation categories Rep(G) of finite groups (codegrees are the degrees of irreducible characters), Tambara‑Yamagami categories, and modular categories arising from quantum groups at roots of unity. In each case the computation confirms that dim 𝒞 and FPdim 𝒞 generate the same Galois‑invariant ideal.

The paper concludes with a discussion of possible extensions. The authors suggest that the formal codegree framework might be adapted to non‑semisimple tensor categories, to infinite‑dimensional settings, or to categories equipped with additional structures such as braidings or pivotal elements. They also point out that understanding the ideal I(𝒞) could shed light on the classification of fusion categories, on the existence of fiber functors, and on the arithmetic of modular data (e.g., the denominators of S‑matrix entries). Overall, the work provides a new algebraic lens through which the numerical invariants of fusion categories can be studied, linking categorical representation theory with classical Galois theory and the arithmetic of algebraic integers.