Splendid Rickard equivalences for 2-blocks with abelian defect groups of order at most 32: the splendid form of Broué's Abelian Defect Group Conjecture in this range

For 2-blocks of finite group algebras whose defect group is an abelian 2-group of order at most 32, Broué's Abelian Defect Group Conjecture is now verified in its plain derived-equivalence form: Ardito–Sambale (2020) settled the elementary abelian case $D\cong C_2^5$ , and Eaton–Livesey's 2024 classification of 2-blocks with abelian defect groups of rank at most four yields the conjecture for every remaining such group. Both works, however, explicitly disclaim the strongest standard form: neither establishes a splendid Rickard equivalence — a tilting complex built from trivial-source bimodules inducing compatible local equivalences and an isotypy — for the blocks it treats, and outside the classical cyclic cases and a few individual principal blocks (for example those of $\mathrm{SL}_2(8)$ and its automorphism group) no splendid equivalence is known across this family. The problem is to verify Broué's conjecture in this splendid form for all 2-blocks with abelian defect groups of order at most 32. Whether the splendid form holds for all such blocks remains open.

Background

Let $(K,\mathcal{O},k)$ be a 2-modular system: $\mathcal{O}$ is a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $2$ and fraction field $K$ of characteristic $0$ . For a finite group $G$ , the group algebra $\mathcal{O}G$ decomposes as a direct sum of blocks, the indecomposable two-sided summands corresponding to the primitive central idempotents; each block $B$ carries a defect group $D$ , a 2-subgroup of $G$ defined up to conjugacy that measures the complexity of $B$ — for instance, $D$ is a Sylow 2-subgroup of $G$ exactly when $B$ is the principal block, the block containing the trivial character. Brauer's first main theorem gives a bijection between the blocks of $G$ with defect group $D$ and the blocks of the normalizer $N_G(D)$ with defect group $D$ ; the block $b$ of $\mathcal{O}N_G(D)$ corresponding to $B$ is its Brauer correspondent. Part of the local structure of $B$ is encoded by its inertial quotient $E=N_G(D,b_D)/C_G(D)$ , an odd-order subgroup of $\operatorname{Aut}(D)$ built from a maximal $B$ -subpair $(D,b_D)$ , i.e. a defect group paired with a block of its centralizer.

Broué's Abelian Defect Group Conjecture (1988) asserts that when $D$ is abelian, $B$ and its Brauer correspondent $b$ are derived equivalent, meaning their bounded derived categories of finitely generated modules are equivalent. The strengthened form due to Rickard predicts more: the equivalence should be a splendid Rickard equivalence — a two-sided tilting complex of $B$ - $b$ -bimodules whose terms are direct sums of direct summands of permutation modules and which induces compatible equivalences between the local categories of Brauer pairs of $B$ and $b$ (a Brauer pair being a pair $(Q,b_Q)$ consisting of a 2-subgroup $Q$ and a block $b_Q$ of $\mathcal{O}C_G(Q)$ ), in particular an isotypy, a canonical correspondence between the ordinary irreducible characters respecting local data. Splendid equivalences retain local information that a plain derived equivalence may lose, and this strongest form is needed for many character-theoretic applications; it is the form in which the conjecture is frequently stated.

The problem sits inside a classification program motivated by Donovan's conjecture, which predicts that there are only finitely many Morita equivalence classes — equivalences of module categories — of blocks with a given defect group. For abelian defect groups at the prime $2$ the Morita equivalence classes are now completely classified, where the rank of an abelian 2-group is the minimal number of elements needed to generate it: rank at most two by Eaton–Kessar–Külshammer–Sambale (2014), rank three by Eaton–Livesey (2018), $D\cong C_2^5$ by Ardito (2021, thirty-four classes), and finally all abelian 2-groups of rank at most four by Eaton–Livesey (2024). Ardito–Sambale (2020) closed the three cases in which Ardito's $C_2^5$ classification had left the inertial quotient undetermined, using the theory of lower defect groups (multiplicities of subgroups as local invariants extracted from the Cartan matrix of a block, the matrix recording the composition of projective modules into simples), and thereby verified the derived form of Broué's conjecture for $D\cong C_2^5$ . Eaton–Livesey (2024) then obtained the derived form for all 2-blocks with abelian defect group of rank at most four, so that Broué's conjecture in the derived form is now known for every 2-block with abelian defect group of order at most $32$ , the cyclic cases being classical. Both papers, however, explicitly record the same gap: Ardito–Sambale state that they do not prove that their derived equivalences are splendid, and Eaton–Livesey state that they “make no claim to the existence of a splendid equivalence” in their result. Whether the splendid form holds across this whole family is the open question recorded here.

Problem Statement

Let $(K,\mathcal{O},k)$ be a 2-modular system with $k$ algebraically closed and $\mathcal{O}$ complete, let $G$ be a finite group, and let $B$ be a 2-block of $\mathcal{O}G$ whose defect group $D$ is an abelian 2-group of order at most $32$ ; let $b$ be the Brauer correspondent of $B$ in $\mathcal{O}N_G(D)$ . Determine whether $B$ and $b$ are splendidly Rickard equivalent: an affirmative answer exhibits an equivalence of their bounded derived categories implemented by a two-sided tilting complex of $B$ - $b$ -bimodules whose terms are direct sums of direct summands of permutation bimodules — bimodules whose underlying $\mathcal{O}(G\times N_G(D))$ -module admits a basis permuted by the group action — and which induces compatible equivalences between the local categories of Brauer pairs of $B$ and $b$ (a Brauer pair being a pair $(Q,b_Q)$ consisting of a 2-subgroup $Q$ and a block $b_Q$ of $\mathcal{O}C_G(Q)$ ), in particular an isotypy, i.e. a bijection between the ordinary irreducible characters of $B$ and $b$ compatible with the local data attached to each character. The problem asks for such a determination for every such block: the task is resolved either by establishing a splendid Rickard equivalence for all such blocks or by exhibiting one such block that admits no splendid Rickard equivalence. The published Morita classifications — the thirty-four classes for $D\cong C_2^5$ classified by Ardito together with the class lists of Eaton–Livesey for the abelian 2-groups of rank at most four, the cyclic cases being classical — organize the task, but since splendidness is not known to transfer along Morita equivalences, treating one representative per class is not by itself sufficient: the question concerns every such block. A valid alternative answer is a refutation: an explicit 2-block with abelian defect group of order at most $32$ that admits no splendid Rickard equivalence with its Brauer correspondent. Plain derived equivalences between $B$ and $b$ are already known for all these blocks; it is precisely the splendidness of the equivalence that is at issue.

References

Cesare Giulio 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 (published as Adv. Group Theory Appl. 12 (2021), 71–78).
Cesare Giulio Ardito, “Morita equivalence classes of blocks with elementary abelian defect groups of order 32”, Journal of Algebra 573 (2021), 297–335, DOI 10.1016/j.jalgebra.2020.12.036, https://doi.org/10.1016/j.jalgebra.2020.12.036.
Charles W. Eaton and Michael Livesey, “Morita equivalence classes of 2-blocks with abelian defect groups of rank 4”, Journal of the London Mathematical Society 109 (2024), Article e12943, DOI 10.1112/jlms.12943; arXiv:2310.05734, https://arxiv.org/abs/2310.05734.
Charles W. Eaton and Michael Livesey, “Classifying blocks with abelian defect groups of rank 3 for the prime 2”, Journal of Algebra 515 (2018), 1–18, DOI 10.1016/j.jalgebra.2018.08.010, https://doi.org/10.1016/j.jalgebra.2018.08.010.
Charles W. Eaton, Radha Kessar, Burkhard Külshammer and Benjamin Sambale, “2-blocks with abelian defect groups”, Advances in Mathematics 254 (2014), 706–735, DOI 10.1016/j.aim.2013.12.024, https://doi.org/10.1016/j.aim.2013.12.024.
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.
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.
Kun Zhang and Yuanyang Zhou, “Inertial 2-blocks with abelian defect groups”, arXiv:2604.09186 (2026), https://arxiv.org/abs/2604.09186.