Splendid Equivalence in Modular Representations

• Splendid equivalence is a refined form of derived and Morita equivalences constructed via p-permutation modules that preserve local data such as defect groups and fusion systems.
• It employs techniques like K-stability, local-to-global gluing, and Brauer indecomposability to establish explicit equivalences between block algebras.
• Advances include classification of splendid Morita equivalence classes, application to Broué’s abelian defect group conjecture, and descent theory for field independence.

Splendid equivalence is a category of derived or Morita equivalences between blocks of group algebras that require the equivalence bimodules or complexes to be built from p-permutation modules, enforcing a fine-grained compatibility with the local structure of finite groups—namely, with defect groups, fusion systems, and source algebras. The notion underpins many advances in modular representation theory, especially through its role in the formulation and verification of Broué’s abelian defect group conjecture and its refinements.

1. Core Definitions and Conceptual Framework

A splendid equivalence is, in the modular representation theory of finite groups, an enhanced equivalence between module categories of block algebras which is realized by bimodules or complexes with strong permutation-theoretic properties. Given blocks $A$ and $B$ (typically of group algebras $kG$ and $kH$ over a field $k$ of characteristic $p$), one says there is a splendid Morita equivalence between $A$ and $B$ if there exists an $(A, B)$-bimodule $M$ that induces an equivalence of module categories and is, viewed as a $k[G \times H]$-module, a $p$-permutation module (i.e., it decomposes into direct summands with vertices of the diagonal form).

A splendid Rickard equivalence is a derived equivalence induced by a bounded complex of $(A, B)$-bimodules, each a $p$-permutation module, such that $C \otimes_B C^* \simeq A$ and $C^* \otimes_A C \simeq B$ in the homotopy category.

The essential ingredient is the use of $p$-permutation modules, guaranteeing that the equivalence interacts well with structural data such as defect groups and the associated fusion systems.

2. Methodologies and Technical Structure

2.1 $K$-Stability and Reduction to Simple Groups

A technical advance is the introduction of $K$-stable complexes, where the group $K$ encodes extra symmetry related to conjugation and the action of normalizers and automorphism groups. For a normal subgroup $H$ of $G$ and an abelian Sylow $p$-subgroup $P$ of $H$, the setup involves

$$K = (H \times H')A(G'),\quad H' = N_H(P),\quad G' = N_G(P)$$

with the requirement that the Rickard complex is $K$-stable, i.e., for all $(x, x') \in H \times H'$, the complex is equivariantly stable. This allows a reduction of Broué's conjecture to the verification for nonabelian simple groups and their automorphism groups, dramatically simplifying the needed cases by introducing a manageable stability condition in lieu of extendibility (Zhou, 2014).

2.2 Local-to-Global and the Gluing Method

Local analysis is fundamental: the role of Scott modules with specified vertices is central, and local equivalences (via Brauer constructions at all subgroups $Q \leq P$) are "glued" to produce global splendid Morita or Rickard equivalences. Brauer indecomposability—indecomposability of the Brauer construction $M(Q)$ for all $Q \leq P$ as a $k[Q\,C_G(Q)]$-module—is a necessary property for the stability of this gluing process and for later lifting to derived equivalence (Koshitani et al., 2020, Koshitani et al., 2022, Koshitani et al., 2019).

2.3 Source Algebra and Cohomological Mackey Functors

Splendid equivalence at the block level descends to equivalence between categories of cohomological Mackey functors, and an equivalence here (particularly in characteristic zero) reflects a unique splendid equivalence at the block level. This is formalized through an intrinsic description: for a block $b$ with defect group $P$ and source algebra $A$, the functor category is encoded in $$E = \operatorname{End}_A(\bigoplus_{Q \leq P} A \otimes_{OQ} O)$$, so a splendid derived equivalence at the source algebra level lifts to an equivalence of cohomological Mackey functor categories (Linckelmann, 2015, Linckelmann et al., 2016).

2.4 Galois Descent and Field of Definition

Descent theory ensures splendid equivalences constructed over a splitting field (or large enough coefficient ring) can, under the criterion of Galois-stability, descend to a smaller base field (possibly even $\mathbb F_p$ for $p$-nilpotent groups). If a chain complex $C$ of modules over an extension $k'$ is Galois-stable (i.e., ${}^\sigma C \simeq C$ for all $\sigma$ in the Galois group), then a unique chain complex exists over $k$ such that $k' \otimes_k \tilde C \simeq C$. This result adapts Reiner's theorem for modules to chain complexes and is indispensable in establishing the base field independence of the existence of splendid Rickard equivalences for $p$-nilpotent groups with abelian Sylow $p$-subgroup (Miller, 25 May 2024, Boltje et al., 2021).

3. Structural and Local Properties

3.1 Brauer Pairs and Local Invariants

A recent technical refinement is the definition of Brauer pairs for splendid Rickard complexes, generalizing the classical notion from $p$-permutation modules. Brauer pairs $(\Delta(P, \varphi, Q), e \otimes f^*)$ link the twisted diagonal of $G \times H$ (arising from an isomorphism $\varphi: Q \rightarrow P$) and block idempotents $e$, $f$ in the corresponding centralizer algebras. Critical is the property that applying the Brauer construction at these pairs yields chain complexes whose non-contractibility signals their relevance for the equivalence. The Lefschetz invariant of the Rickard complex encodes the global $p$-permutation equivalence, and the paper (Breland et al., 2023) establishes that the set of Brauer pairs for the complex coincides with those for the corresponding $p$-permutation equivalence.

3.2 Induced Local Equivalences

Splendid Rickard equivalences not only induce derived equivalences globally but also, through local analysis at these maximal Brauer pairs and their associated normalizers $N_G(P, e)$, yield splendid equivalences between block algebras of normalizers and centralizers. These local considerations are essential in the context of blockwise local-global conjectures and underpin much of the evidence for conjectures such as Broué’s abelian defect group conjecture (Breland et al., 2023).

4. Classification and Finiteness Results

Splendid Morita equivalence classes are, in many cases, fully classified by group-theoretical data:

• Principal $2$-blocks with dihedral, semidihedral, generalized quaternion, or wreathed defect groups are classified up to splendid Morita equivalence, each equivalence realized via an explicit Scott bimodule.
• Puig’s Finiteness Conjecture is established for principal blocks with the above defect groups: for a fixed $p$-group defect group, only finitely many splendid Morita equivalence classes arise (Koshitani et al., 2017, Koshitani et al., 2018, Koshitani et al., 2020, Koshitani et al., 2023).
• These classifications, including exceptional and wild-type cases, depend crucially on embedding results, ambient fusion systems, and detailed local module analysis.

5. Applications, Implications, and Broader Context

Splendid equivalence enables:

• The verification of refined local-global conjectures (Broué, Alperin–McKay, and their extensions) in new families of groups (e.g., alternating groups, general linear groups over finite fields) by showing the existence of splendid Rickard complexes over minimal fields and coefficient rings (Huang, 2021, Huang et al., 2023).
• The derivation of refined numerical invariants and their independence from the choice of splendid equivalence, as in the case of the Broué invariant (Boltje, 2020):

$B(y) = \epsilon(y) \cdot \frac{b(B)}{b(C)}$

where $b(B)$ is computed using local data (Brauer pairs, blocks of centralizers).

• The transfer of deep homological or categorical structure (through tilting complexes, cohomological Mackey functors, or source algebra automorphisms) along the equivalence, with explicit descriptions given for the group of splendid Morita auto-equivalences (Karagüzel et al., 2023).

6. Technical and Methodological Advances

• The methodology for identifying and constructing splendid Rickard equivalences has become increasingly robust, with techniques such as:
  • DG-module and 2-interior algebra formalism for the translation between chain complexes and module categories (Zhou, 2014);
  • Systematic exploitation of fusion systems in controlling the automorphism groups (e.g., splitting the group of splendid Morita auto-equivalences as $$\mathrm{Out}_P(A) \times \mathrm{Out}(P, \mathcal F)$$);
  • The extension and application of the gluing method and stripping-off techniques to produce explicit equivalences in settings of wild representation type.

Summary Table: Local-to-Global Principle in Splendid Equivalence

Step	Input Structure	Output Structure
Local Module Analysis	Brauer pairs, Scott module	Brauer indecomposability, gluing
Splendid Equivalence	Glued complexes/bimodules	Morita/Rickard equivalence of blocks
Local Equivalences	Maximal Brauer pairs	Equivalences of normalizer blocks

This table shows that establishing strong local control (Brauer indecomposability and classification of relevant Brauer pairs) enables the construction of global splendid equivalences whose structure translates down to equivalences for local subalgebras (normalizers, centralizers) as well.

7. Outlook and Current Directions

Current research continues to expand the contexts where splendid equivalence can be constructed, refined, and descended, for instance:

• Extending descent theory to ever larger classes of blocks and base fields (Miller, 25 May 2024),
• Investigating new obstructions and counterexamples illuminating the delicate field-of-definition issues in lifting and descending Rickard complexes,
• Further developing the linkage between the module-theoretic, categorical, and group-theoretic data (fusion systems, source algebra automorphisms),
• Applying refined invariants such as the Broué invariant to analyze and classify derived equivalences in block theory.

Through these developments, splendid equivalence serves as a central organizing principle in the modular representation theory of finite groups, driving the understanding of local-global phenomena, block invariants, and the overall structure of module categories and their links to group theory.