Broué's abelian defect group conjecture and the lifting of isotypies to p-permutation equivalences

The source paper (McHugh 2023) proves that the character-level data carried by a p-permutation equivalence between blocks of finite group algebras — a strong isotypy — is equivalent to the equivalence itself, and frames two open regimes: Broué's abelian defect group conjecture, which predicts that a block with abelian defect group is splendidly Rickard equivalent to its Brauer correspondent, and the question of when an observed Broué isotypy lifts to a p-permutation equivalence. A follow-up by the same author (published in 2026 in the Proceedings of the American Mathematical Society) shows that not every isotypy lifts (Kessar's isotypies between Galois conjugate blocks), via Kühlshammer–Puig class obstructions. The open problem is the conjunction of the still-open general Broué conjecture and the characterization of exactly which isotypies are shadows of module-level equivalences.

Background

Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},k)$ a $p$-modular system — a complete discrete valuation ring $\mathcal{O}$ of characteristic $0$ with algebraically closed residue field $k$ of characteristic $p$ and fraction field $K$ — assumed large enough for $G$. A block is an indecomposable two-sided direct summand of the group algebra $\mathcal{O}G$; the module category of $\mathcal{O}G$ decomposes as a product of the module categories of its blocks, so blocks are the elementary units of modular representation theory. Each block $A$ has a defect group $D$, a $p$-subgroup of $G$ determined up to conjugacy that measures the failure of $A$ to be semisimple; when $D$ is abelian one speaks of an abelian defect. Brauer's main theorems attach to $A$ a Brauer correspondent $B$: a block of the algebra $\mathcal{O}N_G(D)$ of the normalizer of $D$ with the same defect group, whose $k$-theory tightly controls that of $A$.

In 1990 Broué conjectured that when $D$ is abelian this local control has a structural explanation: the bounded derived categories of $A$ and $B$ should be equivalent, and at the character level there should be a perfect isometry — a sign-compatible bijection between the irreducible ordinary characters of $A$ and $B$ satisfying explicit integrality conditions. Rickard (1996) refined this by predicting a splendid Rickard equivalence: an equivalence of derived categories implemented by a chain complex of $(A,B)$-bimodules whose terms are direct summands of permutation modules (so-called $p$-permutation or trivial source bimodules) with suitable self-duality vertices. Boltje and Xu (2008) formalized the Grothendieck-group shadow of this — a $p$-permutation equivalence is a virtual $p$-permutation $(A,B)$-bimodule $\gamma$ whose tensor products with its $\mathcal{O}$-dual satisfy $\gamma\cdot\gamma^{\circ}=[A]$ and $\gamma^{\circ}\cdot\gamma=[B]$ — a notion whose general theory Boltje and Perepelitsky developed further; a splendid Rickard equivalence induces a $p$-permutation equivalence, and a $p$-permutation equivalence induces an isotypy in Broué's sense, the converses not being known in general. An isotypy is defined relative to maximal Brauer pairs and an isomorphism of the associated fusion systems — a Brauer pair is a pair $(P,e)$ with $P$ a $p$-subgroup and $e$ a block idempotent of $kC_G(P)$, and the fusion system is the category encoding conjugacy between such pairs — and consists of a family of perfect isometries between the blocks of the centralizers of corresponding Brauer pairs, compatible under generalized decomposition maps (the maps relating ordinary characters and Brauer characters of elements of $p$-power times $p'$-part). Isotypies between blocks with abelian defect and their Brauer correspondents have been verified in every computed example, and Broué's conjecture — in Rickard's and Boltje–Perepelitsky's strengthened forms — is one of the central open problems of block theory, proved for major families such as the symmetric groups (Chuang–Rouquier, via $\mathfrak{sl}_2$-categorification) but open in general.

The source paper studies exactly what a $p$-permutation equivalence records at the character level. After characterizing the group of trivial source modules (indecomposable direct summands of permutation modules) of a block in terms of Boltje–Carman-style coherent character tuples indexed by Brauer pairs, it introduces the strong isotypy: like a Broué isotypy, a compatible family of virtual characters attached to the Brauer pairs, but defined on the normalizers rather than the centralizers; its main theorem is a bijection between $p$-permutation equivalences and strong isotypies, each strong isotypy restricting to an ordinary isotypy. An isotypy is thus only a partial shadow of the predicted module-level equivalence, and the paper leaves open whether the isotypies that occur in practice lift. A decisive datum here is the Kühlshammer–Puig class, a degree-two cohomology class attached to each centric subgroup of a defect group in the fusion system: $p$-permutation equivalences must preserve these classes, while isotypies need not — the mechanism behind the counterexamples of McHugh's follow-up paper (published 2026), which showed that Kessar's isotypies between Galois conjugate blocks do not always lift, and that isotypic blocks need not be $p$-permutation equivalent at all.

Problem Statement

The problem has two linked components sharing one objective — determining when a character-level block correspondence is the shadow of a module-level equivalence.

(1) Broué's abelian defect group conjecture (standard formulation, in Rickard's strengthened form): let $G$ be a finite group, $p$ a prime, $(K,\mathcal{O},k)$ a $p$-modular system large enough for $G$ with $k$ algebraically closed; let $A$ be a block of $\mathcal{O}G$ with abelian defect group $D$ and let $B$ be the Brauer correspondent of $A$, a block of $\mathcal{O}N_G(D)$. Does there exist a splendid Rickard equivalence between $A$ and $B$? A splendid Rickard equivalence induces a $p$-permutation equivalence, and a $p$-permutation equivalence is equivalent to a strong isotypy (McHugh 2023, Theorem 8.8), but the converses are not known in general, so these are successively weaker formulations of the conjecture. The conjecture is proved for several families (e.g. symmetric groups) and open in general. A proof that every block with abelian defect group admits a splendid Rickard equivalence with its Brauer correspondent resolves it; so does a counterexample — a block with abelian defect group admitting no $p$-permutation equivalence, hence a fortiori no splendid Rickard equivalence, with its Brauer correspondent.

(2) The lifting characterization: determine necessary and sufficient conditions under which an isotypy à la Broué between blocks extends to a strong isotypy, equivalently lifts to a $p$-permutation equivalence. (Whether every $p$-permutation equivalence lifts to a splendid Rickard equivalence is a separate open question, not part of this component.) It is known that such lifting is not automatic — Kessar's isotypies between Galois conjugate blocks do not always lift, and some isotypic blocks admit no $p$-permutation equivalence, the obstruction being mismatched Kühlshammer–Puig classes — but no characterization of the liftable isotypies is known. The abelian-defect instance of this question — whether the isotypies predicted between a block with abelian defect and its Brauer correspondent, verified in all computed examples, always lift — coincides with the $p$-permutation-equivalence formulation of component (1), the intermediate (strictly weaker) level in the hierarchy above rather than the full splendid Rickard equivalence asked there, and is expected to have a positive answer.

References

1. John Revere McHugh, "Coherence conditions for the characters of trivial source modules and strong isotypies", arXiv:2310.10880 (2023), https://doi.org/10.48550/arXiv.2310.10880
2. John McHugh, "Isotypic blocks of finite group algebras that are not p-permutation equivalent", Proceedings of the American Mathematical Society (2026), doi:10.1090/proc/17701, https://doi.org/10.1090/proc/17701 (preprint arXiv:2506.03446)
3. Michel Broué, "Isométries parfaites, types de blocs, catégories dérivées", Astérisque 181–182 (1990), 61–92, Société mathématique de France, https://numdam.org/item/AST_1990__181-182__61_0/
4. Jeremy Rickard, "Splendid equivalences: derived categories and permutation modules", Proceedings of the London Mathematical Society (3) 72 (1996), 331–358, doi:10.1112/plms/s3-72.2.331, https://doi.org/10.1112/plms/s3-72.2.331
5. Robert Boltje and Bangteng Xu, "On p-permutation equivalences: Between Rickard equivalences and isotypies", Transactions of the American Mathematical Society 360 (2008), 5067–5087, doi:10.1090/s0002-9947-08-04393-6, https://doi.org/10.1090/s0002-9947-08-04393-6
6. Robert Boltje and Philipp Perepelitsky, "p-Permutation equivalences between blocks of group algebras", Journal of Algebra 664 (2025), 815–887, doi:10.1016/j.jalgebra.2024.09.038, https://doi.org/10.1016/j.jalgebra.2024.09.038 (preprint arXiv:2007.09253)
7. Radha Kessar, "On Isotypies Between Galois Conjugate Blocks", in Buildings, Finite Geometries and Groups, Springer Proceedings in Mathematics 10, Springer (2012), 153–162, doi:10.1007/978-1-4614-0709-6_7, https://doi.org/10.1007/978-1-4614-0709-6_7
8. David Benson and Radha Kessar, "Blocks inequivalent to their Frobenius twists", Journal of Algebra 315 (2007), 588–599, doi:10.1016/j.jalgebra.2007.03.044, https://doi.org/10.1016/j.jalgebra.2007.03.044
9. Joseph Chuang and Raphaël Rouquier, "Derived equivalences for symmetric groups and sl2-categorification", Annals of Mathematics 167 (2008), 245–298, doi:10.4007/annals.2008.167.245, https://doi.org/10.4007/annals.2008.167.245
10. Robert Boltje and Rob Carman, "The orthogonal unit group of the trivial source ring", arXiv:2203.12718 (2022), https://doi.org/10.48550/arXiv.2203.12718
11. Robert Boltje and John Revere McHugh, "Fixed points of extended tensor products", arXiv:2512.18150 (2025), https://doi.org/10.48550/arXiv.2512.18150