Higher Verlinde Categories Verₚⁿ

• Higher Verlinde categories Verₚⁿ are finite symmetric tensor categories defined as abelian envelopes of tilting module quotients, generalizing fusion categories to settings of positive characteristic.
• Their fusion rules, governed by truncated Chebyshev recurrences and p-adic expansions, reflect intricate block structures and are derived via both classical and quantum Frobenius embeddings.
• These categories categorify cyclotomic integer rings, exhibit incompressibility and subterminality, and play a key role in tensor category theory and Witt group classifications.

Higher Verlinde categories ${\sf Ver}_{p^n}$ are a hierarchy of finite symmetric tensor categories constructed over an algebraically closed field of characteristic $p>0$, by forming abelian envelopes of certain quotients of the category of tilting modules for the algebraic group ${\rm SL}_2$. These categories generalize classical Verlinde fusion categories to positive characteristic and higher nilpotence levels, categorifying cyclotomic integer rings and providing a rich family of incompressible tensor categories that mirror, modulo $p$, many structural aspects of tensor categories at roots of unity in characteristic zero. The higher Verlinde framework also extends to "mixed" and braided settings via Lusztig's quantum groups and quantum Frobenius-Lusztig embeddings, yielding categories ${\sf Ver}_{p^{(n)}}^\zeta$ with intricate block, center, and fusion structure.

1. Construction via Tilting Module Quotients and Abelian Envelopes

Let $k$ be an algebraically closed field of characteristic $p>0$. The rigid Karoubian tensor category ${\rm Tilt} \, {\rm SL}_2$ consists of all direct sums of indecomposable tilting modules $T_m$ for ${\rm SL}_2(k)$, with highest weights $m \in \mathbb{N}$. For each $n \ge 1$, define a thick tensor ideal $I_n$ in ${\rm Tilt} \, {\rm SL}_2$ generated by the $n$-th Steinberg module $St_n := T_{p^n-1}$.

The higher Verlinde category is then defined as

$${\sf Ver}_{p^n} := \text{Abelian Envelope}\left( {\rm Tilt} \, {\rm SL}_2 \, / \, I_n \right)$$

where the abelian envelope is taken in the sense of [Benson-Etingof-Ostrik]. This produces a finite, symmetric, rigid, abelian tensor category with enough projectives and a fully faithful embedding from the quotient category (Benson et al., 2020, Coulembier et al., 2023). The same construction generalizes for arbitrary connected reductive groups $G$ under certain large $p$ assumptions, leading to categories ${\sf Ver}_{p^n}(G)$ as abelian envelopes of suitable quotients of ${\rm Tilt} \, G$ (Newton, 16 Jan 2026).

In the "mixed" or quantum case, over a primitive $N$-th root of unity $\zeta\in k$, tilting modules for the Lusztig quantum group $U_\zeta$ are used, resulting in ribbon, often non-symmetric, categories ${\sf Ver}_{p^{(n)}}^\zeta$ as abelian envelopes of quotients ${\rm Tilt}^\zeta / J_{p^{(n)}}$, where $J_{p^{(n)}}$ consists of tiltings $T_\zeta(m)$ for $m \ge p^{(n)}-1$ (Décoppet, 2024).

2. Simple Objects, Block Structure, and Fusion Rules

The simple objects in ${\sf Ver}_{p^n}$ are indexed by integers $0 \le i < p^{n-1}(p-1)$, corresponding to tilting module labels subject to truncation at the Steinberg level. Each simple $L_i$ may be written as a tensor product of images of lower-level tiltings: $$L_i \cong F(T_{i_1}) \otimes \cdots \otimes F(T_{i_n})$$ where the $i_j$ are the coefficients in the $p$-adic expansion of $i$ with $0 \le i_j < p$, $i_n < p-1$ (Benson et al., 2020, Coulembier et al., 2023).

Fusion rules in ${\sf Ver}_{p^n}$ are induced by those in ${\rm Tilt} \, {\rm SL}_2$ but truncated to reflect the ideal quotient. Explicitly, for simples $L_{i}, L_{j}$,

$$L_i \otimes L_j \cong \bigoplus_{k = |i-j|, |i-j|+2, \ldots}^{\min(i+j, 2M - (i+j))} L_k$$

where $M = p^{n-1}(p-1) - 1$, and the direct sum is taken in steps of two, mirroring $su(2)_k$ Verlinde rules at level $p^n-2$ (Coulembier et al., 2023, Benson et al., 2020, Coulembier et al., 2024, Newton, 16 Jan 2026).

In the mixed case ${\sf Ver}_{p^{(n)}}^\zeta$, the decomposition of simple objects follows a Steinberg tensor product formula: every simple is uniquely of the form

$$L_\zeta(a) = \mathbb{T}_\zeta(a_0) \otimes qFL(L(b))$$

where $a = a_0 + \ell b$, $0 \le a_0 < \ell$ and $qFL$ is the quantum Frobenius-Lusztig embedding from a lower-stage category, with corresponding fusion rules structured as Chebyshev–type recursions and truncated Clebsch-Gordan summations (Décoppet, 2024).

Block decomposition in these categories is dictated by $p \ell$–adic expansions. Simples are partitioned into blocks with sizes determined by the multiplicative structure of the $p, \ell$ parameters, and blocks of the same size (across different $n$ or $\zeta$) are equivalent (Décoppet, 2024).

3. Braided, Symmetric, and Ribbon Structures

When constructed from classical tilting modules ($\zeta = \pm 1$), the categories ${\sf Ver}_{p^n}$ are symmetric tensor categories, with the abelian envelope process preserving this property (Benson et al., 2020, Décoppet, 2024). For $\zeta \neq \pm 1$, ${\sf Ver}_{p^{(n)}}^\zeta$ inherits a canonical ribbon (in particular: braided, spherical, pivotal, and twist) structure from Lusztig's quantum group.

Choice of square root $\zeta^{1/2}$ determines the possible braiding, as the $R$-matrix on $\mathbb{T}_\zeta(1)$ satisfies a Kauffman skein relation: $$\beta_{1,1} = \lambda \mathrm{Id} + \mu(\mathrm{coev} \circ \mathrm{ev}), \quad \lambda^2 = \zeta^{\pm 1}, \quad \lambda \mu = 1$$ so the braiding is only well defined up to choices of $\zeta^{1/2}$. For $\zeta = -1$, there are two distinct ribbon structures distinguished by the parity of the Müger center (Décoppet, 2024).

The symmetric (Müger) center of ${\sf Ver}_{p^{(n+1)}}^{\zeta^{1/2}}$ is explicitly determined: when $\zeta$ has odd order it is equivalent to the lower-stage symmetric Verlinde category, whereas for even order the center is the "even" subcategory. This explicit control over centers reflects the braided hierarchy and is fundamental for Witt group computations (Décoppet, 2024).

4. Grothendieck Rings, Categorical Dimensions, and Cyclotomic Categorification

The Grothendieck ring $K_0({\sf Ver}_{p^n})$ is the free abelian group on its simple objects, with ring structure determined by fusion rules. For ${\sf Ver}_{p^{(n)}}^\zeta$, there is a canonical isomorphism

$$K_0({\sf Ver}_{p^{(n)}}^\zeta) \cong \mathbb{Z}[x] / (Q_{p^{(n)}}(x) / Q_{p^{(n-1)}}(x))$$

with $Q_k(x)$ the Chebyshev polynomial of the second kind of degree $k-1$ (Décoppet, 2024). The even subcategory then corresponds to a quotient by the minimal polynomial of $2\cos(2\pi/p^{(n)})$.

Frobenius–Perron and quantum dimensions of simples are governed by cyclotomic evaluations: $$\mathrm{FPdim}(L_i) = \frac{\sin\Bigl( \frac{(i+1)\pi}{p^n} \Bigr)}{\sin\Bigl( \frac{\pi}{p^n} \Bigr)}$$ with $L_i$ indexed as above (Coulembier et al., 2023, Benson et al., 2020, Coulembier et al., 2024). The value of $[L_1]$ in the Grothendieck ring satisfies the Chebyshev relation for $2\cos( 2\pi / p^n )$, implying that ${\sf Ver}_{p^n}$ categorifies the real cyclotomic integer ring $\mathbb{Z}[2\cos(2\pi/p^n)]$.

Fusion in the Grothendieck ring can be described as truncated Chebyshev recurrence, matching truncated $SO(2)$ fusion in one factor and classical ${\sf Ver}_{p^{n-1}}$ fusion in the other (Décoppet, 2024).

5. Filtrations, Embedding Functors, and Tower Structure

There exist canonical fully faithful tensor functors

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

constructed via the Frobenius twist—either classically, or as the quantum Frobenius-Lusztig embedding $qFL$ in the mixed case. For ${\sf Ver}_{p^{(n)}}^\zeta$, $$qFL: {\sf Ver}_{p^{(n-1)}}^\sigma \to {\sf Ver}_{p^{(n)}}^\zeta$$ is a braided/ribbon embedding and matches objects and fusion rules at the level of $\sigma$ and $\zeta$ parameters (Décoppet, 2024).

This yields a nested hierarchy: $${\sf Ver}_p \subset {\sf Ver}_{p^2} \subset \cdots \subset {\sf Ver}_{p^n} \subset \cdots$$ and the colimit $${\sf Ver}_{p^\infty} := \bigcup_{n} {\sf Ver}_{p^n}$$. Every moderate-growth pretannakian tensor category is conjectured to admit a fiber functor to ${\sf Ver}_{p^\infty}$, a positive characteristic analogue of the classical Tannakian theorem (Benson et al., 2020, Coulembier et al., 2023).

6. Incompressibility, Subterminality, and Witt Group Classes

Higher Verlinde categories are maximally nilpotent (MN): for any simple non-unit, the commutative algebra $\mathrm{Sym}(L)$ is finite, and symmetric powers of generators grow minimally. This property implies incompressibility: every tensor functor from ${\sf Ver}_{p^n}$ to another pretannakian is fully faithful; surjective tensor functors are equivalences; and every moderate-growth tensor category surjects onto an incompressible one, conjecturally a subcategory of ${\sf Ver}_{p^\infty}$ (Coulembier et al., 2023).

In turn, they are also subterminal and Bezrukavnikov: any two tensor functors to ${\sf Ver}_{p^n}$ are isomorphic, and the class of tensor categories fibered over ${\sf Ver}_{p^n}$ is closed under quotients. The mixed and even subcategories share these properties when their centers are incompressible (Décoppet, 2024, Benson et al., 2020).

None of the mixed higher Verlinde categories is a Drinfeld center (absolute or relative); hence, they provide nontrivial elements in pertinent Witt groups. Their characteristic zero analogues recover modular data for $SU(2)$ TQFTs (Décoppet, 2024).

7. Alternative Realizations and Generalizations

Alternate models realize ${\sf Ver}_{p^n}$ as monoidal abelian envelopes of certain quotients of $\mathrm{Rep}(C_p^n)$, using a two-dimensional faithful $C_p^n$-module to produce a tensor ideal, with $D/K$ then identified with ${\rm Tilt} \, SL_2 / I_n$ (Coulembier et al., 2024). This situates ${\sf Ver}_{p^n}$ as the abelian envelope of a quotient of the representation category of an elementary abelian $p$-group.

Generalization to reductive groups $G$ leads to categories ${\sf Ver}_{p^n}(G)$, constructed similarly via quotients of tilting module categories using highest weights compatible with the principal $SL_2$-homomorphism. There are canonical restriction functors to ${\sf Ver}_{p^n}(SL_2)$, Frobenius twist functors inducing inclusions, and subquotients corresponding to Serre subcategories in ${\rm Rep}\, SL_2$ (Newton, 16 Jan 2026).

The explicit functorial machinery bridging arbitrary tensor categories and higher Verlinde categories has been developed through families of higher Frobenius (O$_n$) functors, with conjectures stating that exactness of such a functor detects when a category fibers over ${\sf Ver}_{p^n}$ (Coulembier et al., 2024).

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References:

(Décoppet, 2024, Benson et al., 2020, Coulembier et al., 2023, Newton, 16 Jan 2026, Coulembier et al., 2024).