Permutation, stabilization and decomposition

Abstract: Informed by our understanding of the tt-geometry of permutation modules, we investigate the proper definition of the `stable permutation category' of a finite group. Then we prove that this category decomposes over cyclic and generalized quaternion groups and only in those cases.

• The paper introduces the stable permutation category StPerm(G;k) by innovatively localizing permutation modules in a tensor-triangular framework.
• It establishes that the category is indecomposable except when the p-Sylow subgroup is cyclic or generalized quaternion, providing explicit decomposition formulas.
• The methodology leverages tt-geometry and modular fixed-point functors to create a bridge between support theory and classical modular representation theory.

Permutation, Stabilization, and Decomposition: A Technical Summary

Overview and Motivations

This paper investigates the intricate structure of categories arising from permutation modules over group algebras, particularly introducing and analyzing the stable permutation category $\StPerm(G;k)$ for a finite group $G$ and a field $k$ of characteristic $p>0$. The construction of $\StPerm(G;k)$ is analogized with the classical stable module category $\StMod(kG)$ but departs significantly in both formulation and properties, specifically tailored to capture tensor-triangular (tt) phenomena intrinsic to the world of $p$-permutation modules.

A main result is a definitive decomposition theorem for $\StPerm(G;k)$: except when the $p$-Sylow subgroup of $G$ is cyclic or generalized quaternion, this category is indecomposable as a tt-category; in these exceptional cases, explicit decomposition formulas are established.

Figure 1: Artist rendering of $G$0 highlighting the topological stratification of the spectrum in the Klein-four group case.

Definition and Construction of the Stable Permutation Category

Let $G$1 be finite and $G$2 a field of $G$3. The authors build on existing tt-geometry for permutation modules, considering the bounded homotopy category of $G$4-permutation $G$5-modules, denoted $G$6, as the fundamental triangulated setting. The classical stable module category $G$7 is constructed as the Verdier quotient $G$8.

The naive analogue—simply quotienting $G$9-permutation modules by projectives—fails to provide a meaningful triangulated (or even tt-) category. The authors instead define the stable permutation category as

$k$0

where $k$1 is the tt-ideal of equivariantly perfect (eq-perf) complexes: those $k$2 for which, for every $k$3-subgroup $k$4, the complex of modular $k$5-fixed points (via Brauer quotient functors $k$6) is perfect for the Weyl group $k$7.

Passing to Ind-completions yields the 'big' version, $k$8, as a finite localization of $k$9 away from $p>0$0.

Tensor-Triangular Geometric Justification

The construction of $p>0$1 is grounded in tt-geometry. The spectrum $p>0$2 stratifies via conjugacy classes of $p>0$3-subgroups, with each stratum homeomorphic to the homogeneous spectrum of a group cohomology ring, possibly "extended" projective varieties. Unlike the derived or stable module categories, whose spectra are always local, $p>0$4 is semi-local: it contains finitely many closed points, one for each conjugacy class of $p>0$5-subgroups.

Key technical results establish that:

• $p>0$6 consists precisely of objects whose (tt-)support lies in the set of closed points,
• The spectrum of $p>0$7 is the open semi-local space obtained by excising these closed points:

$p>0$8

These observations connect $p>0$9 as the tt-localization of $\StPerm(G;k)$0 onto the "punctured" spectrum, reminiscent of the localization to the projective spectrum in classical and modular representation theory.

Structural and Functorial Properties

$\StPerm(G;k)$1 forms a Mackey 2-functor with respect to restrictions along injective group homomorphisms, inheriting 2-functoriality and supporting well-defined induction, restriction, and modular fixed-points (Brauer quotient) functors between stable permutation categories. For arbitrary $\StPerm(G;k)$2, the spectrum of $\StPerm(G;k)$3 admits a colimit description over all elementary abelian subquotients, analogous to foundational results on the tt-spectrum of permutation module categories.

The modular fixed points and restriction functors—crucial to the stratification theory—preserve eq-perf complexes, facilitating a robust tt-geometry. Furthermore, $\StPerm(G;k)$4 is cohomological in the precise sense of Mackey 2-functors.

The Indecomposability and Decomposition Theorems

A central theorem asserts that $\StPerm(G;k)$5 is indecomposable as a tt-category unless the $\StPerm(G;k)$6-Sylow subgroup of $\StPerm(G;k)$7 is cyclic or generalized quaternion. The spectrum is connected except in these cases. The proof leverages the irreducibility of the spectra for elementary abelian groups and the colimit construction over elementary abelian sections. When the $\StPerm(G;k)$8-Sylow is not of exceptional type, all non-empty opens in the spectrum can be shown to be connected, precluding nontrivial product decompositions.

For cyclic or quaternion $\StPerm(G;k)$9-Sylow, explicit decompositions are established:

• For $\StMod(kG)$0 cyclic $\StMod(kG)$1-group of order $\StMod(kG)$2,

$\StMod(kG)$3

where each component corresponds to tt-localization at a spectrum point associated to the subgroup tower.

• For generalized quaternion $\StMod(kG)$4,

$\StMod(kG)$5

partitioning the stable permutation category into the stable permutation category of the dihedral group $\StMod(kG)$6 and the stable module category.

These decompositions are reflected precisely in the disconnection of the appropriate spectra; they are realized explicitly via (stable) modular fixed-points functors and canonical localization.

Theoretical and Practical Implications

The introduction of $\StMod(kG)$7 as an intermediate "stable" category for permutation modules establishes a new invariant in tt-geometry, sensitive to group-theoretic and cohomological structures in ways not captured by classical stable module theory. The precise indecomposability criterion and explicit decompositions for cyclic and quaternion cases enable finer classification of stable phenomena in modular representation theory.

Practically, these results facilitate refined support-theoretic analysis for permutation and related modules, with implications for equivariant motives, modular representation theory, and potentially equivariant stable homotopy theory (as $\StMod(kG)$8 is shown to correspond with certain module categories over Bredon cohomology spectra).

The conjectured relation between the "periodic locus" of the spectrum and the definition of $\StMod(kG)$9, established for $p$0-groups and anticipated in general, points to deep connections between localization, periodicity, and tensor-triangulated invariants.

Conclusion

This work delivers a comprehensive framework for stable categories arising from permutation modules, giving robust tt-geometric foundations and resolving the criteria for tt-indecomposability. The approach opens further avenues for investigation, including the explicit description of the periodic locus, extensions to more general representations, and potential applications in motivic and equivariant settings. The spectrum-based decomposition results also provide templates for analogous theorems in broader tensor-triangular contexts.