Modular invariants for group-theoretical modular data. I

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.

💡 Research Summary

The paper addresses the classification problem for commutative separable (special Frobenius) algebras inside untwisted group‑theoretical modular categories, i.e. the Drinfeld centers 𝒞(G)=Z(Vec_G) with trivial 3‑cocycle ω=1. Such algebras are precisely the algebraic objects that encode full conformal field theory extensions and give rise to modular invariants. The authors first set up the necessary background on modular categories, group‑theoretical examples, and the notion of a special Frobenius algebra. They then prove that every indecomposable commutative separable algebra A in 𝒞(G) is uniquely determined by a pair (H,β) where H⊂G is a subgroup and β∈H²(H,k^×) is a 2‑cocycle. Concretely, A is the direct sum of simple objects indexed by the elements of H, each equipped with a one‑dimensional projective representation twisted by β. The multiplication, unit, and Frobenius form are expressed in terms of the group law on H and the cocycle β, showing that A is a “Lagrangian‑type” algebra without the need for a non‑trivial associator because ω=1.

Having identified the algebras, the paper proceeds to describe their local modules. A local module is an A‑module whose half‑braiding with A is trivial; this condition forces the underlying object to be supported on the normalizer N_G(H). The authors demonstrate an equivalence between the category of local A‑modules and the category Rep^β(N_G(H)/H) of β‑projective representations of the quotient N_G(H)/H. The simple local modules correspond to pairs (g,ρ) where g runs over representatives of N_G(H)/H and ρ is an irreducible β‑projective representation of the stabilizer of g. This description yields explicit formulas for the quantum dimensions of the modules and for their fusion rules.

With the algebra–module data in hand, the authors construct the associated modular invariant matrix Z. By definition Z_{X,Y}=dim Hom_𝒞(X⊗Y,A). Substituting the (H,β) parametrization gives a closed‑form expression: Z is a block‑diagonal matrix whose blocks are indexed by double cosets of H in G, and each entry is weighted by the values of β. Consequently, every modular invariant for a group‑theoretical modular datum arises from a choice of subgroup H and a 2‑cocycle β, providing a complete classification of such invariants.

The final part of the work tackles the equivalence problem for group‑theoretical modular categories as ribbon categories. The authors introduce the notion of “isocategorical” groups: two finite groups G and G′ are isocategorical if their representation categories Vec_G and Vec_{G′} are equivalent as tensor categories. They prove that Z(Vec_G) and Z(Vec_{G′}) are ribbon‑equivalent if and only if G and G′ are isocategorical and there exists a subgroup K⊂G×G′ together with a 2‑cocycle γ∈H²(K,k^×) that simultaneously realises the (H,β) data for both categories. In other words, the ribbon equivalence classes are parametrized by the same subgroup‑cocycle pairs up to the natural action of Aut(G)×Aut(G′). This result not only recovers known criteria for equivalence of group‑theoretical modular categories but also quantifies the number of distinct ribbon equivalences between any two such categories.

Overall, the paper delivers a thorough algebraic classification: indecomposable commutative separable algebras ↔ (subgroup, 2‑cocycle) pairs, local modules ↔ projective representations of the normalizer quotient, modular invariants ↔ explicit block matrices derived from the same data, and ribbon equivalences ↔ isocategorical group pairs equipped with compatible cocycles. These findings have immediate applications in the construction of full 2‑dimensional topological quantum field theories, the analysis of boundary conditions in rational conformal field theories, and the systematic study of modular invariants beyond the ADE classification.