Functorial Equivalence of Blocks

• Functorial Equivalence of Blocks is a framework that equates indecomposable subcategories using derived, module-theoretic, or functor-category techniques.
• It preserves key invariants, decomposition theories, and homological properties, enabling structural comparisons across diverse algebraic settings.
• Applications span Lie superalgebra category O, group algebras, and Mackey functor categories, reducing complex representation problems to simpler models.

A functorial equivalence of blocks, in modern representation and category theory, refers to an explicit, category-level equivalence—typically realized by derived or module-theoretic functors—between “blocks” (indecomposable summands with specified central and local data) in representation categories. Functoriality emphasizes the preservation and transport of structure, allowing invariants, decomposition theories, and homological properties to be compared or transferred between contexts such as group algebras, Lie superalgebras, Mackey functor categories, or Hecke algebras attached to $p$-adic groups.

1. General Notion and Categorical Realization

A block in an abelian category, such as the category $\mathcal{O}$ for a Lie superalgebra or the module category for a group algebra $kG$, is a full indecomposable subcategory—often defined by a central character, a defect group, or other invariants (e.g., the integral Weyl group in category $\mathcal{O}$ for Lie superalgebras (Cheng et al., 2013)). Functorial equivalence is realized by an exact (or, more generally, triangulated) functor $F: \text{Block}_1 \to \text{Block}_2$ which is an equivalence of categories, typically relative to a symmetric tensor structure or respecting highest weight structures.

Formally, if $A$, $B$ are (finite-dimensional) algebras (e.g., blocks of group algebras), blocks $A$ and $B$ are (Morita) functorially equivalent if there exists a bimodule $M$ such that $M \otimes_A -$ is an equivalence $A$-mod $\to$ $B$-mod. This is further enhanced in the derived setting: $A$ and $B$ are derived equivalent if there exists a two-sided tilting complex inducing an equivalence $D^b(A)$ $\cong$ $D^b(B)$.

In categorical terms, functorial equivalence may also operate in more refined settings, such as via functors between categories of Mackey functors, $p$-local Mackey algebras, or categories of functors derived from block idempotents using explicit functor categories $\mathcal{F}_{Rpp_k}^\Delta$ or its stable variants (Bouc et al., 2022, Bouc et al., 2023).

2. Functorial Equivalence in Category $\mathcal{O}$ for Lie Superalgebras

The BGG category $\mathcal{O}$ for $\mathfrak{gl}(m|n)$ is partitioned into blocks defined by central character $\chi_X$ and the integral Weyl group $W_{(X)}$. In (Cheng et al., 2013), a functorial equivalence is constructed as follows:

• Twisting functors $T_{(a)}$ (for even simple roots $a$) act as explicit right-exact endofunctors of $\mathcal{O}$, shifting highest weights within orbits dictated by the root datum.
• Odd reflections handle the nonuniqueness of Borel subalgebras in the superalgebra setting, transforming highest weight structures for blocks attached to isotropic odd roots.
• Parabolic induction and restriction: After a sequence of functorial twists and reflections, parabolic induction from an "integral" Levi subalgebra $g_{(X)}$ yields a highest weight categorical equivalence

$$\mathrm{Ind}: \mathcal{O}(g_{(X)}) \cong \mathcal{O}_X$$

where the equivalence is implemented at the level of module categories and preserves the highest weight structure.

The upshot: every block in $\mathcal{O}$ for $\mathfrak{gl}(m|n)$ is functorially equivalent (as a highest weight category) to an integral block for a direct sum of general linear Lie superalgebras. The essential formulae are:

• $w \cdot X = w(X + \rho) - \rho$ defines the dot-action;
• $$g_{(X)} = \langle X_\alpha \mid (X+\rho,\alpha^\vee) \in \mathbb{Z} \rangle$$ (integral Levi subalgebra);
• The equivalence is constructed as the composition of twisting, odd reflection, and parabolic induction functors.

Indecomposability of blocks is then established via explicit extension arguments, showing that every pair of simple modules in a block is linked by sequences of nontrivial extensions obtained via the aforementioned functors.

3. Functorial Equivalences in Modular Representation Theory

a. Group Algebras and Mackey Functor Categories

In the representation theory of finite groups, blocks of group algebras $kG$ are central idempotent summands often classified by defect groups and fusion systems. Functorial equivalence arises in several refined settings:

• Diagonal $p$-permutation functors: The key modern tool is the assignment $(G,b) \mapsto RT^\Delta_{G,b}$, the functor representing the block as an object in $\mathcal{F}^\Delta_{Rpp_k}$ (Bouc et al., 2022).
• Functorial equivalence: Two blocks $(G,b)$, $(H,c)$ are functorially equivalent over $R$ (often $R=\mathbb{F}$ of char $0$) if their associated functors are isomorphic in $\mathcal{F}^\Delta_{Rpp_k}$.
• Stable functorial equivalence: Passing to the stable version (modding out projectives, i.e., contributions from the trivial group), provides invariants more sensitive to nonprojective structure, encapsulated in $\overline{\mathcal{F}_{Rpp_k}}$ (Bouc et al., 2023).

Key results assert that functorial equivalence classes are finite for a given defect group (Bouc et al., 2022), that this equivalence preserves many block invariants (number of simple modules, defect group type), and that in particular circumstances (e.g., abelian defect with Frobenius inertia), any block is stably functorially equivalent to a model block such as $(D\rtimes E,1)$, where $D$ is the defect group and $E$ is the inertial quotient acting freely on $D \setminus \{1\}$ (Bouc et al., 2023).

b. Equivalences in Blocks with Specified Defect Groups

Classification theorems for blocks with small defect groups (e.g., cyclic, $V_4$, $Q_8$, $D_8$, $C_2 \times C_4$) show that the functorial equivalence class depends only on the fusion system or inertial quotient of the block (Yılmaz, 2023). In these cases, functorial equivalence coincides with isotypy or splendid Morita equivalence, unifying local-to-global block structure (e.g., the multiplicity formula for the decomposition into simple functors $S_{L,u,V}$ in the semisimple category of diagonal $p$-permutation functors).

Classification results for blocks with larger elementary abelian defect (order $16$ or $32$) (Eaton, 2016, Ardito, 2019) and for generalized cases (e.g., rank four abelian 2-defect groups (Eaton et al., 2023)) show that Morita equivalences—hence, functorial equivalences in module categories—are finite and governed by explicit invariants (such as inertial quotients, Picard groups, and perfect isometries).

4. Refinements: Splendid, Perverse, and Stable Equivalences

• Splendid equivalence (in Rickard's sense): An equivalence realized by a complex of $p$-permutation bimodules, ensuring compatibility with the fusion system and the structure of $p$-permutation modules (Dreyfus-Schmidt, 2014).
• Perverse equivalence: A filtered derived equivalence based on a stratification of simple modules and perversity data, linking global and local derived equivalences via the Brauer functor.
• Stable functorial equivalence: In the quotient by projectives, equivalence is controlled by the isomorphism classes of stable diagonal $p$-permutation functors, parametrized by triples $(L,u,V)$ with $L\neq1$ (Bouc et al., 2023).

These finer equivalences bridge local and global block categories, producing commutative diagrams linking module- and functor-level invariants (e.g., under the Brauer functor in (Dreyfus-Schmidt, 2014)).

5. Explicit Examples and Classification Criteria

• TI Sylow $p$-subgroups: For blocks with TI Sylow $p$-subgroups, the block and its Brauer correspondent in the normalizer are always functorially equivalent, often via stable equivalence of Morita type—without necessarily being perfectly isometric (Yılmaz, 2023).
• Okuyama's elementary equivalence: For blocks of form $F[(C_p\times C_p)\rtimes C_r]$, elementary equivalence is constructed by explicit elementary tilting complexes controlled by subsets $I_0\subset I$ of the simple module indices (Schaps et al., 2 Aug 2024). The catalog of homogeneous maps between the irreducible components is comprehensively described, and the completeness of this catalog for interval $I_0$ underpins the derived (hence, functorial) equivalence.
• Blocks of $p$-local Mackey and cohomological Mackey algebras: Block-level equivalences (derived, Morita, splendid, or permeable) of group algebras lift canonically to equivalences of associated Mackey functor categories (Rognerud, 2013, Rognerud, 2014, Linckelmann, 2015, Linckelmann et al., 2016). Conversely, equivalences of Mackey functor categories imply permeable equivalences of blocks, forming a robust bridge between module-theoretic and functor-categorical block invariants.

6. The Functorial Reformulation of Classical Conjectures

• Alperin’s Weight Conjecture: The conjecture is restated functorially by assigning, to each block $(G,b)$, a functorial invariant in the Grothendieck group of diagonal $p$-permutation functors, namely

$$\operatorname{FAwc}(G, b): \sum_{\sigma} (-1)^{|\sigma|} [G_{(\sigma)}, b_{(\sigma)}]_F = \begin{cases} [S_{1,1,F}] & \text{if } d(b) = 0, \ 0 & \text{if } d(b) > 0. \end{cases}$$

(Boltje et al., 27 Jul 2025) The functorial version is shown to be equivalent to the classical form and, “stably” (i.e., in the stable functor category), always vanishes as $[S_{1,1,F}] = 0$ in the quotient.

Through the functorial paradigm, conjectures and classification theorems are reframed in the setting of categorical invariants, often revealing more transparent and stronger statements, unifying block theory under the language of semisimple functor categories and stable equivalences.

7. Implications and Applications

• Finiteness conjectures (Puig, Donovan): Finiteness of functorial equivalence classes for fixed defect group $D$ follows from explicit functor decompositions in the semisimple category $\mathcal{F}^\Delta_{\mathbb{F}pp_k}$ (Bouc et al., 2022).
• Local-global theory: Functorial equivalence, especially stable and splendid forms, confirms the transferability of key invariants, including decomposition matrices and character-theoretic data, across blocks with the same local structure.
• Descent to more computable or "model" blocks: The structure of equivalence classes, particularly for blocks with well-understood defect groups or local data, allows complex representation-theoretic problems to be reduced to calculations in simpler contexts (principal blocks, semidirect products, or block algebras with explicit crossed product or source algebra structure).

---

In summary, functorial equivalence of blocks is a categorical framework (refined beyond mere Morita or derived equivalence) capturing deep structural symmetries in representation theory. Realized by explicit derived, Rickard, or functor-category-level equivalences—often controlled by local data such as defect group and fusion system—it provides a powerful, finitary method for classifying, comparing, and transferring invariants between blocks. Its reach extends from Lie superalgebra category $\mathcal{O}$ (Cheng et al., 2013) to group algebra blocks, Mackey functor categories, and $p$-adic representation theory, underpinning both theoretical advances and practical applications in contemporary modular representation theory.