Union Verₚ∞(G): Modular Tensor Categories

• Union Verₚ∞(G) is the inductive limit of symmetric tensor categories arising from modular representation theory, uniting finite-level Verlinde categories via Frobenius twists.
• It generalizes constructions for SL₂ by using tilting modules and abelian envelope techniques, ensuring the preservation of tensorial structures and block decompositions.
• The category serves as a universal target for fiber functors from moderate growth tensor categories, linking modular, quantum, and cyclotomic representation theories.

The union ${\sf Ver}_{p^\infty}(G)$ is the inductive limit of a tower of non-semisimple symmetric tensor categories constructed from the modular representation theory of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic $p>0$. Generalizing previous constructions for $SL_2$ and the classical semisimple Verlinde quotients at $p$, these categories are formed via abelian envelopes of suitable quotients of tilting module categories, incorporating structures from the representation theory of $G$ and aligned with deep phenomena in both positive characteristic and quantum topology. The infinite-level fusion, as the union of all ${\sf Ver}_{p^n}(G)$ via Frobenius twist, is designed to serve as a universal symmetric tensor category into which all categories of moderate growth are expected to admit fiber functors. This construction is closely related to categorification of cyclotomic rings, provides organizational frameworks for modular and quantum representation theory, and suggests far-reaching analogies with Deligne’s theorem and Tannakian formalism in characteristic zero (Benson et al., 2020, Newton, 16 Jan 2026).

1. Finite-Level Verlinde Categories and Their Construction

For $G$ a connected reductive group with Coxeter number $h\le p$, select a principal embedding $\phi: SL_2 \to G$. In the category $Tilt\,G$ of tilting $G$-modules, subcategories $T_n(G)$ are defined by placing tight restrictions on the highest weights of indecomposable summands: $$T_n(G) := \left\{ X \in Tilt\,G \mid \text{all indecomposable summands of } X \text{ have highest weight in } \{0\} \cup \bigl((p^{n-1}-1)\rho+\Lambda^+\bigr) \right\}$$ where $\Lambda^+$ is the set of dominant weights for a simply connected cover of $G$, and $\rho$ is the Weyl vector.

Inside $T_n(G)$, a minimal thick tensor ideal $I_n(G)$ is characterized by highest weights outside a shifted fundamental alcove. The quotient $T_n(G) / I_n(G)$ forms the Karoubian datum for the abelian envelope construction, yielding: $${\sf Ver}_{p^n}(G) = C\left( T_n(G) / I_n(G)\,,\, J_n(G) / I_n(G) \right)$$ where $C(-)$ denotes construction of the comodule category for a finite coalgebra associated to the splitting ideal. Alternatively, one may enlarge to certain subcategories $\overline T_n(G)$ of $\Rep\,G$, leading to the equivalent description: $${\sf Ver}_{p^n}(G) \simeq C\left( \overline T_n(G) / \overline I_n(G),\, \overline J_n(G) / \overline I_n(G) \right)$$ The resulting categories are finite, rigid, symmetric tensor categories whose simples and projectives are controlled by precise combinatorics of highest weights. Restriction along $\phi$ canonically supplies tensor functors to ${\sf Ver}_{p^n}(SL_2)$ (Newton, 16 Jan 2026).

2. Inductive Limit: The Union ${\sf Ver}_{p^\infty}(G)$ via Frobenius Tower

The Frobenius twist endofunctor on $\Rep\,G$ preserves the stratification of tilting subcategories and tensor ideals: $$(-)^{(1)}: \overline T_n(G) \to \overline T_{n+1}(G),\qquad (\overline I_n(G))^{(1)} = \overline I_{n+1}(G)$$ Consequently, the system of inclusions

$${\sf Ver}_{p^n}(G) \longrightarrow {\sf Ver}_{p^{n+1}}(G)$$

is fully faithful and compatible with the symmetric tensor structures.

The infinite-level category is then defined as the colimit: $${\sf Ver}_{p^\infty}(G) := \bigcup_{n \ge 1} {\sf Ver}_{p^n}(G) \cong C\left( \overline T_\infty(G) / \overline I_\infty(G) \right)$$ where $G_{\rm perf}$ is the perfection of $G$, and

$$\overline T_\infty(G) = \bigcup_{r \ge 0} \overline T_{r+1}(G)^{(-r)}, \quad \overline I_\infty(G) = \bigcup_{r \ge 0} \overline I_{r+1}(G)^{(-r)}$$

This inductive structure guarantees that projective and simple objects, block decompositions, and all tensorial properties are stable in the inductive limit (Newton, 16 Jan 2026).

3. Structural Properties and Connections to Representation Theory

The categories ${\sf Ver}_{p^n}(G)$ retain a close relationship to $\Rep\,G$:

• Any bounded exact sequence among objects in $\overline T_n(G)$ remains exact upon passage to ${\sf Ver}_{p^n}(G)$.
• Symmetric and exterior power constructions are inherited where defined.
• The limit category ${\sf Ver}_{p^\infty}(G)$, while possessing infinitely many simples (parametrized by $X(T)$, the weight lattice), continues to retain foundational exactness and block structure.

For $G=SL_2$, simple objects correspond to ranges of highest weights; equivalence with a Serre quotient description of subcategories of $\Rep\,SL_2$ is explicitly established. This illuminates how the categories ${\sf Ver}_{p^n}$ and ${\sf Ver}_{p^\infty}(SL_2)$ serve as both examples and organizing templates for more general $G$ (Newton, 16 Jan 2026, Benson et al., 2020).

4. Universal Properties and Fiber Functor Conjectures

Each finite-level category ${\sf Ver}_{p^n}(G)$ is universal for faithful tensor functors out of $T_n(G)/I_n(G)$. The limit category ${\sf Ver}_{p^\infty}(G)$ inherits this role for the union of underlying Karoubian data.

A central conjecture posits that for any symmetric tensor category $\mathcal{C}$ of moderate growth over $k$, there exists a fiber functor

$$F : \mathcal{C} \longrightarrow {\sf Ver}_{p^\infty}$$

Partial results confirm this for Frobenius-exact and fusion categories (with the image in ${\sf Ver}_{p}$ for the latter) (Benson et al., 2020). This extends the paradigm of Tannakian formalism to the positive characteristic, non-semisimple setting, mirroring Deligne's theorem in characteristic zero and suggesting ${\sf Ver}_{p^\infty}(G)$ as the universal recipient for fiber functors from categories of moderate growth.

5. Relation to Quantum Groups, Verlinde Categories, and Categorification

For $G=SL_2$, each ${\sf Ver}_{p^n}$ is the positive-characteristic reduction of a characteristic-zero semisimple Verlinde category associated to quantum $\mathfrak{sl}_2$ at roots of unity. A flat braided deformation over the Witt ring $W(k)[\zeta]$ interpolates between the classical case and modular representation theory.

The Grothendieck ring $K_0({\sf Ver}_{p^n})$ is isomorphic as a ring to $\mathbb{Z}[2\cos(2\pi/p^n)]$, providing an abelian categorification of the real cyclotomic integer rings. This foundational link embeds the structure theory of ${\sf Ver}_{p^n}$ and its union into the domain of cyclotomic and quantum invariants (Benson et al., 2020).

Incorporation of an affine group scheme $G$ enables the definition of categories of $G$-objects internal to ${\sf Ver}_{p^\infty}$, realized as categories of comodules of internal Hopf algebras. This internalization yields “twisted” forms of classical and quantum groups in positive characteristic and extends to rich categorical representation frameworks for $G$ (Benson et al., 2020, Newton, 16 Jan 2026).

6. Problems, Perspectives, and Open Directions

Open problems include the explicit description of Tannakian objects realizing ${\sf Ver}_{p^n}(G)$ and ${\sf Ver}_{p^\infty}(G)$ as representation categories of affine group schemes internal to the respective categories. Another open direction is the systematic interpolation between these positive characteristic categories and higher Verlinde categories for quantum groups at roots of unity, especially to further connect modular representation theory and quantum topology.

A plausible implication is that the inductive limit structure and universality of ${\sf Ver}_{p^\infty}(G)$ may yield new structural insights for tensor categories in positive characteristic, supporting classification efforts and categorical approaches to modular representation theory and generalizations of the Langlands program in non-semisimple contexts (Benson et al., 2020, Newton, 16 Jan 2026).