Detection Theorem for Permutation Modules

• The detection theorem defines explicit criteria using invariants, coinvariants, and cohomological vanishing to confirm permutation module structures.
• It integrates classical approaches by Weiss with modern refinements, linking integral representation theory and homological algebra.
• Refined detection conditions, including subgroup invariants and block analysis, ensure accurate identification of permutation lattices in finite p-group representations.

A detection theorem for permutation modules provides explicit criteria, often expressed in terms of invariants, coinvariants, cohomological vanishing, or diagram-theoretic data, that enable recognition—or “detection”—of permutation module structures in the integral representation theory of finite groups. Such theorems underlie a large segment of the structure theory for group rings and lattice modules over $p$-groups, bridging the domains of integral representation theory, homological algebra, and module-theoretic characterizations.

1. Fundamentals: Permutation Modules and Lattices

Let $G$ be a finite group and $R$ a commutative ring, typically a complete discrete valuation ring (DVR) of mixed characteristic, such as $\mathbb{Z}_p$ or its finite extensions. An $R G$-lattice is a finitely generated, free $R$-module equipped with a linear action of $G$. An $R G$-permutation module is a direct sum of modules of the form $R[G/H]$ for various subgroups $H\leq G$; equivalently, it is an $R G$-lattice admitting an $R$-basis permuted setwise by $G$.

Permutation modules are of central interest due to their tight connection with the category of $G$-sets, the structure of cohomological invariants, and their appearance in the context of Mackey functors, Galois cohomology, and integral representation theory of $p$-groups (MacQuarrie et al., 2018, Torrecillas et al., 2012).

2. Classical Detection Theorems: Weiss and Cohomology

The foundational result in the area is due to Weiss, who established an explicit criterion for permutation modules over finite $p$-groups. Given a complete DVR $R$ and $G$ a finite $p$-group with $N\triangleleft G$, the detection theorem asserts:

Let $U$ be an $R G$-lattice. If $\operatorname{Res}^G_N U$ is a free $R N$-module and $U^N$ (the $N$-fixed points) is a permutation $R[G/N]$-module, then $U$ is a permutation $R G$-module (MacQuarrie et al., 2018).

For cyclic $p$-groups and unramified $p$, Torrecillas and Weigel refined the detection to a purely cohomological criterion via the vanishing of first cohomology:

$$U \ \text{is a permutation } R G\text{-lattice} \iff H^1(U, \operatorname{Res}^G_U M) = 0\ \text{for all } U \leq G$$

More precisely, over a complete DVR $R$ of characteristic zero (with maximal ideal $pR$) and finite cyclic $p$-group $G$, an $R G$-lattice $M$ is a permutation lattice if and only if $H^1(U, M) = 0$ for all $U \leq G$. This is equivalent to all coinvariants $M_U = M / \omega_{R[U]} M$ being $R$-torsion-free (Torrecillas et al., 2012, Estanislau, 27 Oct 2025).

3. Modern Developments: The Refined Detection Theorem

Recent advances generalize and sharpen the detection theorems, providing conditions valid over arbitrary complete DVRs, ramified or unramified, and for general $p$-groups (not just cyclic):

General Recognition Theorem (Estanislau, 24 Nov 2025)

Let $G$ be a finite $p$-group, $N\triangleleft G$, $R$ a complete DVR of mixed characteristic, and $U$ an $R G$-lattice. Then $U$ is a permutation $R G$-module if and only if:

1. $\operatorname{Res}^G_N\,U$ is an $R N$-permutation module;
2. $U^N$ is $G/N$-coflasque (i.e., $H^1(L, U^N) = 0$ for all $L \leq G/N$) and $U_N$ is a permutation $R[G/N]$-module;
3. $(U/U^N)_N$ is a permutation block $R[G/N]$-module (it splits as a sum of $A_i G$-permutation lattices where $A_i = R/p^i R$).

This theorem refines earlier criteria by incorporating “block” structure in the intermediate torsion layers and extending the necessity as well as sufficiency of the conditions. Importantly, when $U|_N$ is free, the result specializes to Weiss’ theorem; for $N$ of order $p$ and $R=\mathbb{Z}_p$, it reduces to the result of MacQuarrie–Zalesskii (Estanislau, 24 Nov 2025, MacQuarrie et al., 2022).

Summary of the key structural invariants:

Invariant	Description	Role in Detection
$\operatorname{Res}^G_N U$	Restriction to $N$	Must be permutation module
$U^N$	$N$-invariants	Must be $G/N$-coflasque
$U_N$	$N$-coinvariants	Must be permutation module
$(U/U^N)_N$	Coinvariants of quotient	Permutation block module

The proof employs induction on $|G|$ and the $R$-rank of $U$, splitting the module along invariant and coinvariant layers, and using vanishing of Ext and cohomology to control summands.

4. Special Cases: Cyclic $p$-Groups and Cohomological Criteria

For cyclic $p$-groups, the detection problem admits further simplification. When $p$ is unramified in $R$, the equivalence reduces to:

$$U \text{ is a permutation } R G\text{-lattice} \iff H^1(L, U) = 0 \ \forall L \leq G$$

This is the full coflasque criterion. If $p=2$ or $R$ is unramified, every coflasque lattice is permutation (Estanislau, 27 Oct 2025). For ramified $R$, coflasque implies permutation only over $C_p$, and the detection theorem still applies provided invariants and coinvariants are checked at a subgroup of order $p$.

These results, originally due to Butler–Reiner, Weiss, and extended by Torrecillas–Weigel and Estanislau, allow explicit deduction of permutation structure from cohomological data or coinvariants, subject to careful attention in ramified settings.

5. Diagrammatic and Butler Correspondence Methods

Butler introduced a correspondence translating the module structure problem to combinatorial data—diagrams—over subquotients:

• For $G=C_p \times C_p$ and $R=\mathbb{Z}_p$, Butler’s method encodes lattices as tuples $(V; V_{(i)})$ of $F_p G$-modules and submodules, converting the detection criteria to diagram-chasing conditions.
• The MacQuarrie–Zalesskii theorem (MacQuarrie et al., 2022) shows—for $N$ of order $p$ and $R = \mathbb{Z}_p$—that $U$ is permutation if and only if the $N$-invariants, $N$-coinvariants, and $(U/U^N)_N$ all satisfy diagram-theoretic (permutation) conditions.
• Importantly, counterexamples constructed via the diagram method show that checking only invariants and coinvariants is insufficient for $p$ odd; the intermediate block quotient $(U/U^N)_N$ must also be controlled.

This diagram category perspective underpins the proof techniques for non-cyclic groups and provides a way to classify obstructions to permutativity.

6. Derived and Singularity Category Invariants

Recent work conceptualizes the detection problem in the language of derived categories and singularity invariants (Balmer et al., 2020). For a finite group $G$ over a commutative Noetherian ring $R$, the detection functor

$x_H(X) = \operatorname{sing}_R\bigl(X^{hH}\bigr)$

(where $X^{hH}$ denotes derived cohomology) plays a key role. The detection theorem asserts that a bounded complex $X$ of $R G$-modules lies in the thick subcategory generated by permutation modules if and only if $x_H(X)=0$ for all $H\leq G$. In this way, permutation structure is detected at the level of vanishing of singularities in all subgroup cohomologies.

For $p$-groups and suitable $R$, this approach recovers the classical detection theorem and illuminates deeper structural connections in triangulated and singularity categories.

7. Limitations, Counterexamples, and Extensions

The necessity of all detection conditions is essential. Explicit counterexamples for $G = C_p \times C_p$, $p$ odd, constructed using Butler diagrams, show that $U^N$ and $U_N$ being permutation modules does not guarantee $U$ is permutation—there exist reduced lattices where the block quotient fails permutativity, invalidating naive detection (MacQuarrie et al., 2022).

Subtlety arises in the ramified case and for non-cyclic $p$-groups. Not all coflasque lattices are permutation, and the equivalence of criteria may fail outside the specified hypotheses. However, recent theorems (e.g., (Estanislau, 24 Nov 2025, Estanislau, 27 Oct 2025)) provide criteria robust to these obstructions, as long as all invariants, coinvariants, and block conditions are verified.

A pervasive theme in the literature is the reduction of structural recognition to local, computable invariants—augmented by modern categorical tools and explicit module-theoretic calculations. The detection theorem for permutation modules remains a cornerstone in the paper and classification of integral representations of finite groups.