The strengthened Broué abelian defect group conjecture: splendid Rickard equivalence over arbitrary complete discrete valuation rings

Broué's abelian defect group conjecture predicts that a block algebra of a finite group with an abelian defect group is derived equivalent to its Brauer correspondent. Kessar and Linckelmann proposed a strengthened (refined) version: over any $p$ -modular system, with no splitting hypothesis on the coefficient ring, every block with abelian defect group should be splendidly Rickard equivalent — the strongest standard categorical local-to-global equivalence — to its Brauer correspondent in the normalizer of the defect group. The literature lineage, including the recent verification of the conjecture for all blocks of ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ by Huang–Li–Zhang, is as follows: the strengthened conjecture is confirmed for blocks with cyclic or Klein four defect groups, alternating groups, ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ , unipotent blocks of ${\rm GL}_n(q)$ , blocks of $p$ -nilpotent groups with abelian Sylow $p$ -subgroups, and RoCK blocks of double covers of symmetric and alternating groups. Whether the strengthened conjecture holds for every block of every finite group with abelian defect group over every complete discrete valuation ring remains open; the problem asks for a resolution of this general statement.

Background

Let $p$ be a prime. Modular representation theory of finite groups works over a $p$ -modular system $(K,\mathcal{O},k)$ : $\mathcal{O}$ is a complete discrete valuation ring (a principal ideal domain, complete for the topology of its maximal ideal, such as the ring $\mathbb{Z}_p$ of $p$ -adic integers) with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p$ . For a finite group $G$ , the group algebra $\mathcal{O}G$ decomposes as a product of indecomposable two-sided ideals $\mathcal{O}Gb$ , indexed by the primitive idempotents $b$ of the center $Z(\mathcal{O}G)$ ; each $\mathcal{O}Gb$ is a block algebra of $G$ . A defect group $P$ of a block $b$ is a maximal $p$ -subgroup of $G$ on which the Brauer map (the algebra homomorphism from the $P$ -fixed points of $\mathcal{O}G$ to $kC_G(P)$ obtained by reducing coefficients modulo the maximal ideal and keeping only elements commuting with $P$ ) does not annihilate $b$ . By Brauer's first main theorem, the Brauer map gives a bijection between the blocks of $\mathcal{O}G$ with defect group $P$ and the blocks of $\mathcal{O}N_G(P)$ with defect group $P$ , where $N_G(P)$ is the normalizer of $P$ ; the corresponding block of $\mathcal{O}N_G(P)$ is the Brauer correspondent of $b$ . A Rickard equivalence between symmetric algebras $A$ and $B$ is a bounded complex $X$ of finitely generated $(A,B)$ -bimodules, projective on both sides, such that $X\otimes_B X^{*}$ is isomorphic to $A$ plus a contractible complex of $(A,A)$ -bimodules and $X^{*}\otimes_A X$ is isomorphic to $B$ plus a contractible complex of $(B,B)$ -bimodules, where $X^{*}$ is the $\mathcal{O}$ -dual complex; it induces an equivalence of derived module categories. When $A$ and $B$ are block algebras, the Rickard equivalence is called splendid if every term of $X$ is a $p$ -permutation bimodule (a direct summand of a permutation bimodule) whose indecomposable direct summands are relatively projective with respect to the diagonal subgroup $\Delta P=\{(g,g^{-1})\mid g\in P\}$ , that is, their vertices are contained in conjugates of $\Delta P$ ; a splendid Rickard equivalence is the strongest standard local-to-global equivalence between a block and its Brauer correspondent and induces, in particular, a derived equivalence and a perfect isometry between the irreducible characters. Broué's abelian defect group conjecture (1990) predicts that when the defect group $P$ is abelian, $\mathcal{O}Gb$ and its Brauer correspondent are derived equivalent, in Broué's original formulation over a splitting $p$ -modular system (one whose residue field is large enough that all irreducible modular representations of the groups considered are absolutely irreducible). Kessar and Linckelmann later proposed a refined, strengthened version: over any $p$ -modular system, with no splitting hypothesis at all, there should exist a splendid Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}N_G(P)c$ . The strengthening is substantial for two reasons. First, splendid Rickard equivalence is a far more rigid notion than derived equivalence, encoding the local $p$ -subgroup structure of the block. Second, removing the splitting hypothesis creates an arithmetic obstruction: the standard construction strategy first builds an equivalence over a splitting field and then attempts Galois descent to the given coefficient ring, and the known descent criteria require Galois-stability conditions on the terms of the complex that need not hold for an arbitrary group. The conjecture is known in several families: Kessar and Linckelmann proved that Rouquier's splendid Rickard complex for blocks with cyclic defect groups descends to arbitrary $p$ -modular systems; Huang achieved the analogous descent for Klein four defect groups; Huang proved the refined conjecture for all blocks of alternating groups; Huang–Li–Zhang proved it for all $p$ -blocks of ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ and subsequently for the unipotent blocks of ${\rm GL}_n(q)$ ; Miller verified it for blocks of $p$ -nilpotent groups (groups with a normal $p$ -complement) with abelian Sylow $p$ -subgroups; and Du–Huang verified it for RoCK blocks (Rouquier–Chuang–Kessar blocks, model blocks realizing arbitrary abelian defect) of the double covers of symmetric and alternating groups over $\mathbb{Z}_p$ . Broué's original conjecture already remains open for arbitrary finite groups — for instance for general non-cyclic abelian defect groups — and the strengthened conjecture implies it, so the strengthened statement is at least as hard. No restriction to any particular family of groups, defect groups, or coefficient rings is part of the conjecture; its general case, especially blocks with non-cyclic abelian defect groups and arbitrary inertial quotients (the quotient $N_G(P,b)/PC_G(P)$ , which measures the non-locality of the block), is the unresolved target.

Problem Statement

For every prime $p$ , every $p$ -modular system $(K,\mathcal{O},k)$ — equivalently, every complete discrete valuation ring $\mathcal{O}$ with residue field of characteristic $p$ — every finite group $G$ , and every block $b$ of $\mathcal{O}G$ whose defect group $P$ is abelian, with Brauer correspondent $c$ in $\mathcal{O}N_G(P)$ : does there exist a splendid Rickard complex of $(\mathcal{O}Gb,\mathcal{O}N_G(P)c)$ -bimodules, that is, a splendid Rickard equivalence between the block algebra $\mathcal{O}Gb$ and its Brauer correspondent $\mathcal{O}N_G(P)c$ ? This is the strengthened Broué abelian defect group conjecture as stated by Kessar and Linckelmann. The question carries no restriction on the finite group $G$ , on the rank or isomorphism type of the abelian defect group, on the inertial quotient, or on the coefficient ring beyond being a complete discrete valuation ring; a negative answer may be given by an explicit block for which no such splendid Rickard complex exists.

References

Xin Huang, Pengcheng Li, Jiping Zhang, The strengthened Broué abelian defect group conjecture for ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ , arXiv:2111.11145 (2021), DOI 10.48550/arXiv.2111.11145, published in Journal of Algebra 633 (2023) 114–137, DOI 10.1016/j.jalgebra.2023.05.049, https://arxiv.org/abs/2111.11145
Radha Kessar, Markus Linckelmann, Descent of equivalences and character bijections, Springer Proceedings in Mathematics and Statistics 242 (2018) 181–212, DOI 10.1007/978-3-319-94033-5_7; arXiv:1705.07227, https://arxiv.org/abs/1705.07227
Xin Huang, Descent of equivalences for blocks with Klein four defect groups, arXiv:2106.03809 (2021), DOI 10.48550/arXiv.2106.03809; Journal of Algebra 614 (2023) 898–905, DOI 10.1016/j.jalgebra.2022.09.026, https://arxiv.org/abs/2106.03809
Xin Huang, Descent of splendid Rickard equivalences in alternating groups, arXiv:2111.10922 (2021), DOI 10.48550/arXiv.2111.10922; Journal of Algebra 658 (2024) 277–293, DOI 10.1016/j.jalgebra.2024.05.048, https://arxiv.org/abs/2111.10922
Xin Huang, Pengcheng Li, Jiping Zhang, Descent of splendid Rickard equivalences in ${\rm GL}_n(q)$ , arXiv:2302.12129 (2023), DOI 10.48550/arXiv.2302.12129; Journal of Pure and Applied Algebra 228 (2024), article 107711, DOI 10.1016/j.jpaa.2024.107711, https://arxiv.org/abs/2302.12129
Sam K. Miller, Galois descent of splendid Rickard equivalences for blocks of $p$ -nilpotent groups, arXiv:2405.16061 (2024), DOI 10.48550/arXiv.2405.16061; Proceedings of the American Mathematical Society 153 (2025), DOI 10.1090/proc/17230, https://arxiv.org/abs/2405.16061
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Yucong Du, Xin Huang, On RoCK blocks of double covers of symmetric and alternating groups and the refined Broué conjecture, arXiv:2510.02147 (2025), DOI 10.48550/arXiv.2510.02147, https://arxiv.org/abs/2510.02147
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