Tensor Categories Verₚⁿ(G) in Representation Theory

• Tensor Categories Verₚⁿ(G) are finite tensor categories in positive characteristic, defined via quotient constructions of tilting modules and their abelian envelopes.
• They generalize semisimple Verlinde categories by incorporating higher Frobenius functors and rich tensor-categorical structures for a connected reductive group G.
• Explicit constructions, especially for SL₂, reveal Serre quotients and truncated fusion rules that connect tilting theory with modular representation and quantum invariants.

Tensor categories ${\sf Ver}_{p^n}(G)$ are finite tensor categories in characteristic $p$ associated to a connected reductive algebraic group $G$ and a positive integer $n$. They generalize the semisimple Verlinde categories ${\sf Ver}_p(G)$ introduced by Gelfand–Kazhdan and the higher Verlinde categories ${\sf Ver}_{p^n}$ for ${\rm SL}_2$ constructed by Benson–Etingof–Ostrik. The construction of ${\sf Ver}_{p^n}(G)$ is based on a quotient of the category of tilting modules for $G$ and the formation of its abelian envelope. These categories inherit rich tensor-categorical structures and functorialities, and can be explicitly realized as Serre quotients in the case $G={\rm SL}_2$. The union over all $n$ leads to a universal category governed by the perfection of $G$.

1. Construction via Tilting Modules and Abelian Envelopes

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a connected reductive algebraic group with Coxeter number $h$ such that $p \geq \max(h, 2h-4)$ (ensuring Donkin's tensor-product theorem applies). The key ingredients in the construction are:

• The category ${\sf Tilt\,}G \subset {\sf Rep\,}G$ of indecomposable tilting modules.
• Collections of tilting modules parametrized by highest weights $\lambda \in X(T)^+$ (with $T$ a maximal torus and $\Lambda$ the weight lattice):
  • $J_n(G)$: $T(\lambda)$ with $\lambda \in (p^{n-1} - 1)\rho + \Lambda^+$
  • $I_n(G)$: summands in $J_n(G)$ whose Steinberg factor $\mu$ does not lie in the fundamental alcove $\mathcal{A}$
  • $T_n(G)$: spanned by weights in $\{0\} \cup ((p^{n-1} - 1)\rho + \Lambda^+)$
• The inclusions $I_n \subset J_n \subset T_n$ are thick tensor ideals, with $J_n$ minimal above $I_n$; there is a unique maximal tensor ideal $I_n^{\max}$ associated to $I_n$.

The quotient $(T_n/I_n)$ forms a pseudo-tensor category. The abelian envelope of $T_n/I_n$ with respect to the ideal $J_n/I_n$ is realized as the comodules over the coalgebra

$$C = \bigoplus_{T,S \in J_n \setminus I_n} \operatorname{Hom}_{T_n}(T,S)^*,$$

with tensor structure inherited from $T_n$. The resulting category is denoted

${\sf Ver}_{p^n}(G) = C(T_n/I_n, J_n/I_n).$

By construction, it is a finite tensor category; each $T(\lambda) \in T_n$ yields a projective object in ${\sf Ver}_{p^n}(G)$ if $\lambda \in J_n$, or zero otherwise (Newton, 16 Jan 2026).

2. Tensor Structures, Functoriality, and Inclusions

The structure of ${\sf Ver}_{p^n}(G)$ is governed by the Steinberg and Donkin tensor-product theorems:

• $$L(\lambda+p^n \mu) \cong L(\lambda) \otimes L(\mu)^{(n)}$$, for $\lambda < p^n$
• $$T(\lambda+p^n \mu) \cong T(\lambda) \otimes T(\mu)^{(n)}$$, for $\lambda \in (p^n - 1)\rho + \Lambda^+$

Pullback along a principal homomorphism $\varphi:{\rm SL}_2 \to G$ defines tensor functors

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

and preserves the quotient construction at the level of tensor ideals, with $I_n(G) = \varphi^{-1}(I_n({\rm SL}_2))$ (Newton, 16 Jan 2026).

There exist fully faithful Frobenius–twist inclusions

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

compatible with the $(-)^{(1)}$ operation in ${\sf Rep\,}G$. For $p > h$, a further decomposition as

$${\sf Ver}_{p^n}(G) \cong {\sf Ver}^+_{p^n}(G) \boxtimes {\sf Rep}_{\mathrm{sVec}}(Z(G), z)$$

separates the root lattice part and the invertible central part. Taking $n \to \infty$ over the perfection $G_{\rm perf}$, one obtains the union

$${\sf Ver}_{p^\infty}(G) = \bigcup_{n \geq 1} {\sf Ver}_{p^n}(G)$$

corresponding to the abelian envelope of an ascending chain of enlarged categories $\overline{T}_n$ inside ${\sf Rep}(G_{\rm perf})$ (Newton, 16 Jan 2026).

3. Objects, Exact Sequences, and Projectives

In ${\sf Ver}_{p^n}(G)$, the indecomposable projective objects are the images of $T(\lambda)$ for $\lambda$ in $$((p^{n-1}-1)\rho + \Lambda_{n-1} + p^{n-1}\mathcal{A}) \cap X(T)$$. The simple objects are the images of $L(\lambda)$ for $$\lambda \in (\Lambda_{n-1} + p^{n-1} \mathcal{A}) \cap X(T)$$. The category is quasi-hereditary, and the simple and projective objects are classified by their highest weights as inherited from the tilting module theory.

The category ${\sf Ver}_{p^n}(G)$ preserves bounded exactness: all bounded exact sequences in the enlarged tilting subcategory $\overline{T}_n \subset {\sf Rep\,}G$ remain exact after passage to ${\sf Ver}_{p^n}(G)$. This enables explicit construction of exact sequences characterizing symmetric and exterior powers within these tensor categories (Newton, 16 Jan 2026).

4. Specialization to ${\rm SL}_2$, Serre Quotients, and Explicit Descriptions

For $G = {\rm SL}_2$, the construction specializes to previous results:

• The categories $T_n, I_n, J_n$ coincide with those analyzed by Benson–Etingof–Ostrik for non-semisimple higher Verlinde categories ${\sf Ver}_{p^n}$.
• Every thick tensor ideal in ${\sf Tilt\,}{\rm SL}_2$ is among the $I_m$, with $J_n = I_{n-1}$.
• An enlarged construction $\overline{T}_n$ inside ${\sf Rep}{\rm SL}_2$ yields a quotient functor $\overline{T}_n \to {\sf Ver}_{p^n}$ that is essentially surjective on objects.

The abelian category underlying ${\sf Ver}_{p^n} = {\sf Ver}_{p^n}({\rm SL}_2)$ has two concrete descriptions:

1. As a Serre quotient of the finite-length category $A_n$ of ${\rm SL}_2$-modules with highest weights $< p^n-1$ by the Serre subcategory $B_n$ generated by simple modules $L_i$ for $(p-1)p^{n-1} \leq i < p^n-1$. ${\sf Ver}_{p^n} \simeq A_n/B_n$.
2. As the abelian subcategory $C_n \subset A_n$ consisting of all modules $X$ with trivial hom-space to and from $B_n$: $$\operatorname{Hom}(X, B) = \operatorname{Hom}(B, X) = 0$$ for all $B \in B_n$ (Newton, 16 Jan 2026).

This provides a concrete module-theoretic realization: ${\sf Ver}_{p^n}({\rm SL}_2)$ comprises “small” weight modules with no composition factors in a “forbidden” window, with morphisms similarly truncated.

5. Relationship to Higher Frobenius Functors and Classification

The higher Verlinde categories ${\sf Ver}_{p^n}$ can be constructed as the abelian envelope of a quotient of ${\sf Rep}\,E$ for $E \cong (\mathbb{Z}/p)^n$ (elementary abelian $p$-group) by its minimal nonzero tensor ideal. This approach yields:

• An equivalence of pre-tensor categories:

$({\sf Tilt\,}{\rm SL}_2)/I_n \cong D/K$

where $D$ is the subcategory generated by a faithful $2$-dimensional representation $V$ of $E$, and $K$ is the smallest nonzero tensor ideal in $D$ (Coulembier et al., 2024).

Higher Frobenius functors (“$O_n$-functors”) are conjectured to detect fibering over ${\sf Ver}_{p^n}$: for any tensor category $\mathcal{C}$ of moderate growth,

$$\mathcal{C} \text{ admits a tensor functor to } {\sf Ver}_{p^n} \iff \Phi_\mathcal{C} \text{ is exact}.$$

This categorical characterization plays a central role in classifying all tensor categories of moderate growth in characteristic $p$, analogous to the classical Frobenius functor for $n = 1$ (Coulembier et al., 2024).

6. Fusion, Braiding, and Pivotal Structures

The fusion rules in ${\sf Ver}_{p^n}$ (for $G={\rm SL}_2$) are truncations of the classical Clebsch–Gordan rules:

• For $i+j < p^n-1$,

$$L_i \otimes L_j \cong \bigoplus_{k=|i-j|,\,|i-j|+2,\, \ldots,\, i+j} L_k, \quad (k < p^n-1).$$

The unique symmetric braiding and canonical pivotal structures on ${\sf Ver}_{p^n}$ descend from those on ${\sf Rep\,}{\rm SL}_2$ and the Temperley–Lieb category. Projective objects correspond to the block ${\sf Tilt}^{[n]}{\rm SL}_2$ within the specified weight window and their fusion can be computed using tilting module decompositions.

This explicit, functorial structure supports the application of ${\sf Ver}_{p^n}(G)$ in tensor category theory, particularly in positive characteristic, and establishes a bridge between tilting representation theory and categorical tensor-invariant theory (Newton, 16 Jan 2026, Coulembier et al., 2024).

7. Significance and Generalizations

The framework of tensor categories ${\sf Ver}_{p^n}(G)$:

• Unifies classical semisimple Verlinde categories and higher non-semisimple generalizations.
• Connects representation theory of reductive groups, modular representation theory, and categorical structures relevant to quantum type invariants in characteristic $p$.
• Admits concrete module-theoretic realizations and quotient-descriptions, enabling explicit computations in examples, especially for $G={\rm SL}_2$.
• Provides the foundation for higher Frobenius functorialities crucial to the modern understanding and classification of symmetric tensor categories in positive characteristic.

A plausible implication is that the extension to perfection and higher levels $n\to\infty$ organizes all extensions of tilting theory and their quotients in a universal tensor-categorical setting, suggesting a broad unifying structure underlying representation theory in characteristic $p$ (Newton, 16 Jan 2026, Coulembier et al., 2024).