Monoidal Ringel Duality

• Monoidal Ringel duality is a framework that generalizes classical highest weight category dualities by preserving tensor products via compatible monoidal structures.
• It employs tilting objects, presheaf categories, and Day convolution to construct equivalences that transport duality across complex representation theoretic settings.
• Applications span diagrammatic categories, affine Lie algebras, and quantum groups, offering canonical monoidal abelian envelopes for non-semisimple contexts.

Monoidal Ringel duality generalizes classical Ringel duality for highest weight categories by incorporating compatible monoidal structures, yielding equivalences that preserve tensor products. This concept not only unifies phenomena such as tilting and Koszul duality in representation theory, but also enables the transport of tensor category structures through highest weight dualities, producing canonical monoidal abelian envelopes for broad classes of non-semisimple and diagrammatic monoidal categories.

1. Formalism of Monoidal Ringel Duality

Let $\mathcal{C}$ be a $k$-linear abelian highest weight category with weight poset $(\Lambda, \leq)$, standard objects $\Delta(\lambda)$, and costandard objects $\nabla(\lambda)$ for each $\lambda \in \Lambda$. If $\mathcal{C}$ has a compatible tensor structure $(\mathcal{C}, \otimes, 1)$ and its tilting subcategory $\mathrm{Tilt}(\mathcal{C})$ is a rigid monoidal subcategory, monoidal Ringel duality constructs a dual category $\mathcal{E}$ such that:

• $\mathcal{E}$ is again a highest weight category (possibly with reversed poset).
• There is a unique (up to monoidal equivalence) monoidal structure $(\otimes^R, 1^R)$ on a completion of $\mathcal{E}$ (typically denoted $\mathcal{E}^{\mathrm{cc}}$).
• There is a strong monoidal functor

$$R := \operatorname{Hom}_{\mathcal{C}}(T, -) : \mathcal{C} \longrightarrow \mathcal{E}^{\mathrm{cc}}$$

where $T$ is a tilting generator in $\mathcal{C}$ and $$A^R = \operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$$, $\mathcal{E} = A^R\text{-mod}$ (Flake et al., 22 Dec 2025).

The analogs for categories not necessarily finite are constructed via Brundan–Stroppel's semi-infinite Ringel duality, ind/pro-completions, and Day convolution monoidal structures on presheaf categories. In the strict polynomial functor case, Ringel duality arises as the restriction of an invertible monoidal self-equivalence (Krause, 2012).

2. Self-Duality, Ext Symmetries, and Monoidal Structure

For the category $H = C_{\Lambda_n}$ of Kraśkiewicz–Pragacz (KP) modules over the upper-triangular Lie algebra $\mathfrak{b}$, the following structure is established (Watanabe, 2015):

• $H$ is highest weight with poset $\Lambda_n$ (of Lehmer codes for $S_n$) and standard objects $\Delta(\lambda) = S_\lambda$.
• There exists a tilting generator $T = \bigoplus_{\lambda} T(\lambda)$ such that $R := \mathrm{Hom}_{\mathfrak{b}}(-, T): H \to H$ is an exact self-equivalence.
• $R$ is a strong monoidal functor: For all modules with $\Delta$-filtration,

$$(R(M) \otimes R(N))_{\Lambda_n} \cong R \left( (M \otimes N)_{\Lambda_n} \right)$$

where the $\Lambda_n$-quotient refers to restriction to weight spaces in $\Lambda_n$.

• $R$ interchanges standard and costandard objects up to a simple involutive formula, and induces symmetrical isomorphisms on Ext-groups.

These properties endow $R$ with the structure of a monoidal auto-duality, preserving the tensor product in the quotient category.

3. Constructions: Tilting Objects, Presheaves, and Day Convolution

Monoidal Ringel duality utilizes several categorical constructions:

• Tilting Objects: Objects possessing both standard and costandard filtrations. In finite or lower-finite settings, every highest weight admits an indecomposable tilting object; the direct sum of these forms a tilting generator $T$ (Flake et al., 22 Dec 2025).
• Presheaf Categories and Day Convolution: For the Ringel dual category $\mathcal{E}$, the ind-completion $\mathcal{E}^{\mathrm{cc}}$ is canonically equivalent to a presheaf category $\mathrm{PSh}(\mathrm{Tilt}(\mathcal{C}))$ endowed with a monoidal structure via Day convolution:

$$(F \otimes^R G)(T) = \int^{T_1, T_2} \operatorname{Hom}(T, T_1 \otimes T_2) \otimes_k F(T_1) \otimes_k G(T_2)$$

This ensures that the image of $R$ preserves monoidal products, with representable presheaves embedding monoidally.

• Adjunctions and Mutual Equivalence: The left adjoint $L$ (given by $L(M) = T \otimes_{A^R} M$) and $R$ mediate equivalences on tilting and projective subcategories (Flake et al., 22 Dec 2025).

4. Examples and Applications

Monoidal Ringel duality applies to various algebraic and diagrammatic contexts:

• Triangular Categories and Diagrammatics: For Sam–Snowden triangular categories, Day convolution on finite dimensional presheaves yields upper-finite highest weight categories, and monoidal Ringel duality recovers Deligne interpolation categories as tiltings in symmetric monoidal abelian envelopes (Flake et al., 22 Dec 2025).
• Knop's Tensor Envelopes: For regular Mal’cev categories with degree function $\delta$, the Karoubi envelope is symmetric monoidal, and monoidal Ringel duality places classical tensor categories (like $\operatorname{Rep}(S_t)$) as tiltings (Flake et al., 22 Dec 2025).
• Positive-Level Affine Lie Algebras: For category $\mathcal{O}_\kappa$ of affine Lie algebras at $\kappa > 0$, monoidal Ringel duality realizes a braided monoidal structure on the positive level side, extending Kazhdan–Lusztig equivalences and showing that tiltings correspond to quantum group representations (Flake et al., 22 Dec 2025).
• Strict Polynomial Functors and Schur Algebras: In strict polynomial functor categories, Ringel duality corresponds to twisting by a distinguished invertible monoid (the divided-power functor) in the larger monoidal category, restricting to the characteristic tilting functor on Schur algebras. The self-equivalence is realized via monoidal Koszul duality, while Serre duality appears as the square of this autoequivalence (Krause, 2012).

5. Key Commutative Diagrams and Naturality

The compatibility of the monoidal structure with Ringel duality is encoded in naturality squares:

• Functoriality in Variables: Commutative diagrams for exact sequences with $\Delta$-filtration ensure the structure maps $\varphi_{M,N}$ are surjective and isomorphisms once the base cases are verified (Watanabe, 2015).
• Compatibility with Extensions: These diagrams guarantee that surjectivity in monoidality is transferred recursively throughout the subcategory of $\Delta$-filtered objects.
• Monoidal Uniqueness: The universal property of Day convolution ensures the uniqueness (up to monoidal equivalence) of the monoidal structure on the Ringel dual when prescribed by the image of tilting objects and the functor $R$ (Flake et al., 22 Dec 2025).

6. Unification, Broader Significance, and Explicit Low-Rank Cases

Monoidal Ringel duality unifies several concepts in modern representation theory:

• Synthesis: Koszul duality in strict polynomial functors, Ringel duality for Schur algebras, and Serre duality are all monoidal phenomena grounded in the invertibility of a distinguished object (e.g., the divided-power functor) (Krause, 2012).
• Concrete Low-Rank Examples: In the KP-module setting, for $n=2$, Ringel duality reduces to the identity functor, illustrating the trivial autoequivalence. For $n=3$, $R$ swaps certain KP modules, and the translation of filtered exact sequences under $R$ matches dual tensor product decompositions, exemplifying the nontrivial monoidal behavior (Watanabe, 2015).
• Embedding of Diagrammatic and Quantum Categories: Monoidal Ringel duality provides abelian envelopes for diagrammatic and quantum symmetric tensor categories, extending to interpolation categories, Brauer–type diagrammatics, and fusion categories from conformal field theory.

A plausible implication is that monoidal Ringel duality furnishes a general mechanism for constructing and rigidifying tensor categorical structures in contexts where classical quasi-hereditary approaches are insufficient, particularly in infinite or diagrammatically presented settings.

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Key References:

• "Kraśkiewicz-Pragacz modules and Ringel duality" (Watanabe, 2015)
• "Monoidal Ringel duality and monoidal highest weight envelopes" (Flake et al., 22 Dec 2025)
• "Koszul, Ringel, and Serre duality for strict polynomial functors" (Krause, 2012)