Broué's Abelian Defect Group Conjecture in Full Generality

Broué's abelian defect group conjecture (1990) predicts that if $A$ is a block of the group algebra of a finite group $G$ over a $p$ -modular system, and the defect group $D$ of $A$ is abelian, then the bounded derived category of $A$ is equivalent, as a triangulated category, to that of its Brauer correspondent $B$ in the group algebra of the normalizer $N_G(D)$ . The conjecture is confirmed for blocks with cyclic defect groups, for all blocks of symmetric groups, and for many further classes (small abelian defect groups, various finite groups of Lie type, $p$ -solvable groups, and, as of April 2026, blocks with a single irreducible Brauer character orbit or with inertial quotient of prime order — the inertial quotient being the group of automorphisms of the defect group induced by its normalizer in the group), yet it remains open for arbitrary finite groups and arbitrary abelian defect groups; a survey updated in April 2026 states it "remains open in general". The problem recorded here is the full-generality statement: prove it for all blocks with abelian defect groups, or produce an explicit counterexample.

Background

Modular representation theory studies a finite group $G$ over a coefficient system adapted to a prime $p$ dividing $|G|$ : a $p$ -modular system $(K,\mathcal{O},k)$ consisting of a complete discrete valuation ring $\mathcal{O}$ with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p$ (one may work directly over $k$ ). Maschke's theorem fails in characteristic $p$ , and the group algebra decomposes as a direct product of indecomposable two-sided algebras called blocks. Each block $A$ of $\mathcal{O}G$ has a defect group $D$ : a $p$ -subgroup of $G$ , well defined up to conjugacy, which plays the role of a Sylow $p$ -subgroup for the block and measures the failure of projectivity of its modules. Brauer's first main theorem pairs $A$ with a distinguished block $B$ of $\mathcal{O}N_G(D)$ , also with defect group $D$ , its Brauer correspondent (for the principal block, $B$ is the principal block of $N_G(D)$ ).

The invariant conjectured to be shared is the bounded derived category $D^{b}(\mathrm{mod}\,A)$ , the triangulated category obtained from finitely generated $A$ -modules by formally inverting quasi-isomorphisms of chain complexes. Rickard's derived Morita theory shows that two finite-dimensional algebras have equivalent bounded derived categories exactly when one is the endomorphism algebra of a tilting complex over the other (a tilting complex being a bounded complex of finitely generated projective modules whose endomorphism complex has homology concentrated in degree zero); derived equivalence preserves numerical invariants such as the number of simple modules and the Cartan matrix — the matrix recording the composition multiplicities of each simple module in the indecomposable projective modules — up to unimodular equivalence. Observing that in every computed example a block with abelian defect group and its Brauer correspondent admit matching character-level data — a perfect isometry, meaning a bijection between the irreducible ordinary characters of the two blocks compatible with the reduction of character values modulo the maximal ideal of $\mathcal{O}$ , and more strongly a family of compatible perfect isometries called an isotypy — Broué conjectured in 1990 a structural explanation: the two derived categories should be equivalent. Rickard strengthened the expectation to a splendid equivalence, a derived equivalence induced by a tilting complex of permutation bimodules satisfying local conditions, and such equivalences induce isotypies; all verified instances in fact realize the strengthened form.

The conjecture is a theorem in major special cases: for blocks with cyclic defect groups (Rickard, with refinements by Rouquier); for all blocks of symmetric groups and of finite general linear groups in non-defining characteristic, by Chuang and Rouquier's categorification of $\mathfrak{sl}_{2}$ — a lifting of the generators of the Lie algebra $\mathfrak{sl}_{2}$ to functors on module categories, from which the required equivalences are obtained — which, with Chuang–Kessar's determination of the structure of the RoCK blocks (a class of symmetric-group blocks with abelian defect groups and especially regular combinatorial structure), settled the symmetric group case; and, more recently and partially, for many groups of Lie type, for elementary abelian 2-defect groups of order $32$ , for $p$ -solvable groups (groups whose non-abelian composition factors all have order prime to $p$ ), for $\mathrm{SL}(2,p^{n})$ and $\mathrm{GL}(2,p^{n})$ over an arbitrary complete discrete valuation ring, and for blocks with a single irreducible Brauer character orbit or with inertial quotient of prime order (April 2026), the inertial quotient of a block with defect group $D$ being the group of automorphisms of $D$ induced by its normalizer in $G$ . Craven and Rouquier's perverse equivalences — derived equivalences compatible with specified filtrations of the module categories, computable in favorable cases — give one of the main systematic methods. Despite this progress over thirty-five years, no proof or counterexample is known in general; the survey of Jasso–Krause–Schroll (updated April 2026) states that the conjecture "remains open in general", and current research continues to verify it class by class.

Problem Statement

Prove or disprove Broué's abelian defect group conjecture in full generality. Let $G$ be an arbitrary finite group, $p$ a prime, $(K,\mathcal{O},k)$ a $p$ -modular system large enough for $G$ (that is, $K$ and $k$ are splitting fields for $G$ and for all of its subgroups), and let $A$ be a block of $\mathcal{O}G$ whose defect group $D$ is abelian. Let $B$ be the Brauer correspondent of $A$ , the block of $\mathcal{O}N_{G}(D)$ paired with $A$ by Brauer's first main theorem. The conjecture asserts an equivalence of triangulated categories

D^{b}(\mathrm{mod}\,A)\ \simeq\ D^{b}(\mathrm{mod}\,B).

The equivalence is sought in full generality: for every finite group, every prime $p$ , and every block with abelian defect group, with no restriction on the group, the defect group, the block, or the coefficient conventions beyond the standard $p$ -modular system; since a proof of the stronger splendid-equivalence form implies the conjecture, such a proof settles the problem as well. An explicit counterexample may be exhibited at the residue-field level (blocks of $kG$ and $kN_{G}(D)$ ), since a derived equivalence over $\mathcal{O}$ induces one over $k$ ; likewise, a proof of the splendid-equivalence form over $k$ settles the problem, since splendid equivalences over $k$ lift uniquely to $\mathcal{O}$ (Rickard). A plain derived-equivalence proof over $k$ is not interchangeable with the $\mathcal{O}$ -version: the cited literature establishes no general theorem lifting derived equivalences from $k$ to $\mathcal{O}$ , this upgrading being precisely the descent obstruction studied by Kessar and Linckelmann ('Descent of Equivalences and Character Bijections', reference 14).

References

Michel Broué, 'Isométries parfaites, types de blocs, catégories dérivées', Astérisque 181–182 (1990), 61–92. Record: https://numdam.org/item/AST_1990__181-182__61_0/
Jeremy Rickard, 'Derived equivalences as derived functors', Journal of the London Mathematical Society (2) 43 (1991), 37–48. DOI: 10.1112/jlms/s2-43.1.37, https://doi.org/10.1112/jlms/s2-43.1.37
Raphaël Rouquier, 'From stable equivalences to Rickard equivalences for blocks with cyclic defect', in Groups '93 Galway/St Andrews, London Mathematical Society Lecture Note Series, Cambridge University Press, 1995. DOI: 10.1017/cbo9780511629297.017, https://doi.org/10.1017/cbo9780511629297.017
Joseph Chuang and Raphaël Rouquier, 'Derived equivalences for symmetric groups and sl2-categorification', Annals of Mathematics (2) 167 (2008), 245–298. DOI: 10.4007/annals.2008.167.245, https://doi.org/10.4007/annals.2008.167.245
David A. Craven and Raphaël Rouquier, 'Perverse equivalences and Broué's conjecture', Advances in Mathematics 248 (2013), 1–58. DOI: 10.1016/j.aim.2013.07.010, https://doi.org/10.1016/j.aim.2013.07.010
Stefano Sannella, 'An algorithmic approach to perverse derived equivalences: Broué's Conjecture for $\Omega^{+}_{8}(2)$ ', arXiv:1810.01467 (2018). https://arxiv.org/abs/1810.01467
Cesare G. Ardito and Benjamin Sambale, 'Broué's Conjecture for 2-blocks with elementary abelian defect groups of order 32', arXiv:2011.06793 (2020). https://arxiv.org/abs/2011.06793
Xin Huang, Pengcheng Li, and Jiping Zhang, 'The strengthened Broué abelian defect group conjecture for $\mathrm{SL}(2,p^{n})$ and $\mathrm{GL}(2,p^{n})$ ', arXiv:2111.11145 (2021). https://arxiv.org/abs/2111.11145
John Revere McHugh, 'Coherence conditions for the characters of trivial source modules and strong isotypies', arXiv:2310.10880 (2023). https://arxiv.org/abs/2310.10880
Sam K. Miller, 'On endosplit $p$ -permutation resolutions and Broué's conjecture for $p$ -solvable groups', arXiv:2408.04094 (2024). https://arxiv.org/abs/2408.04094
Gustavo Jasso, Henning Krause, and Sibylle Schroll, 'Rickard's Derived Morita Theory: Review and Outlook', arXiv:2509.06369 (2025; v2 updated April 2026). https://arxiv.org/abs/2509.06369
Fuming Jiang, Kun Zhang, and Yuanyang Zhou, 'Blocks with only one irreducible Brauer character orbit', arXiv:2604.09161 (2026). https://arxiv.org/abs/2604.09161
Qianhu Zhou and Kun Zhang, '2-blocks with abelian defect groups and inertial quotient of prime order', arXiv:2604.09974 (2026). https://arxiv.org/abs/2604.09974
Radha Kessar and Markus Linckelmann, 'Descent of Equivalences and Character Bijections', in Geometric and Topological Aspects of the Representation Theory of Finite Groups, Springer Proceedings in Mathematics & Statistics, Springer, 2018, 181–212. DOI: 10.1007/978-3-319-94033-5_7, https://doi.org/10.1007/978-3-319-94033-5_7