Endopermutation Module Insights

• Endopermutation modules are modules over group rings of finite p-groups whose endomorphism algebras admit bases permuted by group elements, forming a permutation structure.
• They are classified via the Dade group up to one-dimensional twists and utilize slash functors to parametrize indecomposable modules in block theory.
• Their fusion stability and twisted diagonal vertex properties enable the construction of stable Morita equivalences between block algebras.

An endopermutation module is a module over the group ring of a finite $p$-group whose endomorphism algebra, under conjugation by group elements, exhibits a permutation module structure. This property makes endopermutation modules foundational in the local theory of block algebras, the structure of interior algebras, and stable Morita equivalences. They are classified up to tensoring with one-dimensional modules by the Dade group, and enable powerful functorial constructions such as slash (deflation–restriction) functors, which are instrumental in parametrising indecomposable modules and proving equivalences between blocks of group algebras.

1. Formal Definition and Equivalent Characterizations

Let $P$ be a finite $p$-group and $O$ a complete discrete valuation ring with residue field $k$ of characteristic $p$. An $OP$-module $V$ is called an endopermutation module if its endomorphism algebra $\End_O(V)$ is $O$-free and admits a basis permuted by the action of $P$ (Biland, 2013, Huang, 30 May 2024). Equivalently,

• $\End_O(V)$ is an $O$-free $(P \times P)$-permutation module (via left-right multiplication).
• Writing $\Delta P = \{(x,x) \mid x \in P\}$, the restriction of $\End_O(V)$ to $\Delta P$ is a permutation $OP$-module with a $P$-stable basis.

Dade's alternative characterisation states that $V$ is endopermutation if $\End_O(V) \cong V \otimes_O V^*$ is a permutation module for $P$ under the diagonal action, so $V \otimes_O V^*$ admits an $O$-basis permuted by $P$ (Linckelmann, 2014). The ordinary character $\chi_V$ of $V$ restricts to integer values on $P$ exactly when the trace of a group element on its fixed-point space is $\pm 1$ (Huang, 30 May 2024).

2. The Dade Group and Classification

Indecomposable endopermutation modules for $P$ are classified by the Dade group $D_O(P)$: the set of classes of such modules under tensor product modulo permutation modules (Linckelmann, 2014, Huang, 30 May 2024). Every indecomposable endopermutation module $V$ factors as

$[V] = \prod_{Q < P} [\Omega_O(P/Q)]^{n_Q}$

for integers $n_Q$, where $\Omega_O(P/Q)$ is the augmentation kernel for the permutation module on cosets $P/Q$. Up to tensoring with one-dimensional modules (linear characters), $V$ can be written as

$$V \cong \chi \otimes \bigotimes_{Q < P} \Omega_O(P/Q)^{\otimes n_Q}$$

(Huang, 30 May 2024). This decomposition yields explicit generators for $D_O(P)$, and establishes an injective map from $D_O(P)$ to the product of Dade groups for factor groups $N_P(Q)/Q$ as $Q$ varies over proper subgroups.

3. Fusion Stability and Block-theoretic Sources

A source module $V$ is said to be fusion-stable with respect to a saturated fusion category if its isomorphism class is invariant under all fusion system morphisms. For block-theoretic applications, the structure of vertices and sources is further refined:

• An indecomposable module over a block algebra $OGb$ has a vertex subpair $(P, e_P)$ and a source triple $(P, e_P, V)$, where $e_P$ is a block idempotent of $OC_G(P)$ and $V$ is an indecomposable endopermutation $OP$-module (Biland, 2013, Huang, 30 May 2024).
• Fusion-stable sources are required for equivalences involving non-principal blocks, with compatibility conditions for restriction maps in the Brauer category of a block.

4. Slash Functors and Functorial Classification

Slash functors generalise the Brauer functor, providing a functorial mechanism to pass from modules over $OGe$ with fusion-stable endopermutation sources to $kH e_P$-modules for subgroups $P \le G$ and intermediate groups $H$ with $P C_G(P) \le H \le N_G(P, e_P)$ (Biland, 2013). The slash functor

$$\text{Sl}_{(P, e_P)} : OGeM \rightarrow kH e_P \text{Mod}$$

preserves additive, tensor, and exact structures, and behaves naturally under conjugation and transitivity of e-subpairs.

The parametrisation theorem asserts a bijection: | Indecomposable $OGe$-modules with source triple $(P, e_P, V)$ | Projective indecomposable $k N_G(P, e_P) e_P$-modules | | ------------------------------------------------------------- | -------------------------------------------------- | | $M \mapsto \text{Sl}_{(P, e_P)}(M)$ | | This holds precisely when the classification of fusion-stable endopermutation modules $V$ is complete, enabling Dade-style classification for Brauer-friendly modules and all endo-$p$-permutation modules in principal blocks (Biland, 2013).

5. Endopermutation Sources and Stable Equivalences

Stable equivalence of Morita type between blocks is often constructed via bimodules with endopermutation sources. Let $A$ and $B$ be source algebras for two $p$-blocks sharing defect group $P$ and fusion system $\mathcal{F}$. For an $\mathcal{F}$-stable endopermutation module $V$, form

$U = A \otimes_{OP} \Ind_{\Delta P}^{P \times P}(V) \otimes_{OP} B$

and select an indecomposable summand $M$. If for each nontrivial fully $\mathcal{F}$-centralized $Q < P$ the canonical local bimodule $M_Q$ gives Morita equivalence of local blocks, then $M, M^*$ induce a global stable equivalence of Morita type (Linckelmann, 2014). The trivial source case recovers classical results due to Alperin, Broué, and Puig; the nontrivial endopermutation source case is enabled by fusion stability and the more general block-theoretic construction.

Endopermutation modules thus serve as universal sources for local-to-global Morita equivalence principles in block theory, as applied to questions such as the $Z_p^*$-Theorem (Linckelmann, 2014).

6. Twisted Diagonal Vertices and Puig’s Theorem

If a stable equivalence bimodule between two block algebras has a twisted diagonal vertex, any source of the bimodule must be an endopermutation module (Huang, 7 Dec 2025). Specifically, for blocks $OGb$ and $OHc$, a bimodule $M$ with vertex $\Delta\varphi \subseteq G \times H$ (for $\varphi: P \to Q$ an isomorphism) yields a source $V$ that is necessarily endopermutation. This follows from Puig's interior-algebra criterion, where the endomorphism algebra of the source embeds as an interior $P$-algebra in a manner necessitating a permutation module structure of $\End_O(V)$, and is preserved under descent to non-algebraically-closed fields under mild conditions. The classification in terms of the Dade group further clarifies the local structure of stable equivalences with twisted diagonal vertices.

7. Isotypies and Applications in Block Theory

Endopermutation modules precisely underpin the construction of Morita equivalences and induce almost isotypies between blocks via slash functors (Huang, 30 May 2024). For blocks $OGb$ and $OHc$ with common defect $P$ and fusion system, an indecomposable bimodule $M$ with fusion-stable endopermutation source $V$ and determinant $1$ ensures:

• Weak isotypy if the character values $\chi_V(u)$ are integers.
• Almost isotypy if the family of slashed local Morita equivalences is compatible up to signs $\varepsilon_{Q,u} = \pm 1$ derived from $V$ and its slashes.
• Under suitable hypotheses (e.g. $p \ge 3$, $P$ abelian), full isotypy as defined by Linckelmann is obtained.

These isotypy structures ensure compatibility of decomposition maps and modular character correspondences, and highlight the necessity of allowing sign anomalies in almost isotypies for certain groups (e.g. quaternion groups) (Huang, 30 May 2024).

Table: Endopermutation Module Properties

Property	Construction/Definition	Reference
Permutation of Endomorphism Ring	$\End_O(V)$ has $O$-basis permuted by $P$	(Biland, 2013)
Dade Group Classification	$[V] = \prod_{Q < P} [\Omega_O(P/Q)]^{n_Q}$	(Huang, 30 May 2024)
Fusion Stability	Invariance of source under fusion system morphisms	(Linckelmann, 2014)
Slash Functor Action	$Sl_{(P,e_P)}(M) \to k N_G(P,e_P) e_P$-module	(Biland, 2013)
Stable Morita Equivalence	Bimodule with endopermutation source induces equivalence	(Linckelmann, 2014)
Twisted Diagonal Vertex Sources	Any source is an endopermutation module	(Huang, 7 Dec 2025)

Endopermutation modules, through their algebraic and categorical properties, provide a comprehensive local framework for understanding the modular representation theory of finite groups, block equivalences, and character-theoretic structures in algebraic settings.