Broué's abelian defect group conjecture over a complete discrete valuation ring for spin blocks of double covers of symmetric groups

Broué's abelian defect group conjecture predicts that a block of a finite group with abelian defect group is derived equivalent to its Brauer correspondent. For the spin blocks of the double covers of the symmetric and alternating groups the conjecture is settled over algebraically closed fields of odd characteristic, and over a complete discrete valuation ring for the special RoCK blocks (Kleshchev–Livesey; descended to $\mathbb{Z}_p$ by Du–Huang). The open problem is the integral case for arbitrary spin blocks: for every odd prime $p$ , every weight $0<d<p$ , and every $\bar{p}$ -core $\rho$ , establish (or refute) a derived equivalence over $\mathcal{O}$ between the spin block $B^{\rho,d}$ (and its even part) and its Brauer correspondent, the expected strong form of which, matching the RoCK case, is a splendid derived equivalence.

Background

Let $p$ be an odd prime and $(\mathbb{K},\mathcal{O},\mathbb{F})$ a $p$ -modular system: $\mathbb{K}$ is a field of characteristic zero, $\mathcal{O}$ a complete discrete valuation ring with fraction field $\mathbb{K}$ and algebraically closed residue field $\mathbb{F}$ of characteristic $p$ . For a finite group $G$ , a block is an indecomposable direct summand (a central-primitive idempotent component) of the group algebra $\mathcal{O}G$ ; its defect group $D$ is a $p$ -subgroup of $G$ controlling its modular representation theory. Brauer's first main theorem states that $G$ has a block with defect group $D$ if and only if the normalizer $N_G(D)$ does, and that the two blocks correspond: the block of $N_G(D)$ is the Brauer correspondent of the block of $G$ . Broué's abelian defect group conjecture (1990) asserts that when $D$ is abelian, the block $B$ of $G$ and its Brauer correspondent $b$ are derived equivalent, i.e. their bounded derived categories of modules are equivalent as triangulated categories; the conjecture can be stated over $\mathbb{F}$ or over $\mathcal{O}$ , and the $\mathcal{O}$ -version is stronger. A splendid derived equivalence is one induced by a bounded complex of bimodules whose terms are direct sums of direct summands of permutation bimodules — equivalently, indecomposable bimodules with endopermutation sources and suitable vertices — an equivalence that also transports the local $p$ -permutation structure (the vertex of an indecomposable module is a minimal subgroup to which the module is relatively projective, and its source is the indecomposable module over the vertex from which it is induced), and which is the form in which integral Broué-type equivalences are usually sought.

Let $\mathsf{S}_n$ be the symmetric group and $\tilde{\mathsf{S}}_n$ one of its two Schur double covers (central extensions of $\mathsf{S}_n$ by a cyclic group of order two), with central involution $z$ ; $\mathbb{K}\tilde{\mathsf{S}}_n(1+z)/2$ is isomorphic to $\mathbb{K}\mathsf{S}_n$ , so the genuinely new part is the spin part $\tilde{\mathsf{S}}_n(1-z)/2$ , whose blocks are the spin blocks, labeled by pairs $(\rho,d)$ where $\rho$ is a $\bar{p}$ -core (the core of a partition under the bar-removal operation governing spin characters) and $d\geq 0$ is the weight, with $n=|\rho|+dp$ . For $d>0$ the defect group of the spin block $B^{\rho,d}$ is abelian exactly when $d<p$ , in which case it is isomorphic to $(C_p)^{\times d}$ . The even part $B^{\rho,d}_{\bar{0}}$ is a spin block of the double cover $\tilde{\mathsf{A}}_n$ of the alternating group.

For symmetric groups themselves the conjecture was settled in two steps. Chuang and Kessar (2002) constructed the RoCK blocks for $\mathsf{S}_n$ — the blocks whose $p$ -core is a Rouquier core (a core with a long, well-separated rim making the block combinatorially tractable) — and proved them Morita equivalent to explicit 'local' wreath-product algebras, giving Broué's conjecture for these blocks. Chuang and Rouquier (2008) then proved, via $\mathfrak{sl}_2$ -categorification, that any two blocks of symmetric groups of equal weight are derived equivalent, which reduces an arbitrary block to a RoCK block and completes Broué's conjecture for all blocks of symmetric groups (over both $\mathbb{F}$ and $\mathcal{O}$ , since the Chuang–Rouquier equivalences are defined over $\mathbb{Z}$ ).

For the double covers the analogous program is newer. Kleshchev and Livesey (Memoirs AMS, 2025) constructed RoCK blocks for the double covers over $\mathbb{F}$ using quiver Hecke superalgebras (KLR-type superalgebras categorifying quantum pairs, related to spin blocks through the Kang–Kashiwara–Tsuchioka isomorphism) and proved Broué's conjecture for them over $\mathbb{F}$ . Ebert–Lauda–Vera (2022, still a preprint) and Brundan–Kleshchev (published in Selecta Mathematica, 2026 issue) independently built the odd analog of Chuang–Rouquier: an odd $\mathfrak{sl}_2$ -categorification acting on spin blocks, showing that any spin block of $\mathbb{F}\tilde{\mathsf{S}}_n$ of weight $d$ is derived equivalent to a RoCK block, thereby completing Broué's conjecture over $\mathbb{F}$ for all spin blocks. Kleshchev and Livesey (Annals of Representation Theory, 2025) then treated the RoCK blocks over $\mathcal{O}$ itself by purely finite-group-theoretic methods: $B^{\rho,d}$ is Morita superequivalent (a Morita equivalence respecting the $\mathbb{Z}_2$ -grading of the superalgebras) over $\mathcal{O}$ to the local model $B^{\rho,0}\otimes(B^{\varnothing,1}\wr_s\mathcal{T}_d)$ , and consequently $B^{\rho,d}$ and $B^{\rho,d}_{\bar{0}}$ are splendidly derived equivalent to their Brauer correspondents over $\mathcal{O}$ . Du and Huang (2025) descended these equivalences to the ring $\mathbb{Z}_p$ of $p$ -adic integers, verifying the Kessar–Linckelmann refined Broué conjecture for the spin RoCK blocks. What no work has achieved is the passage over $\mathcal{O}$ from an arbitrary spin block to a RoCK block: the odd categorification machinery exists only over fields, so the reduction step available over $\mathbb{F}$ has no integral counterpart. This is the acknowledged remaining gap, stated explicitly by Kleshchev–Livesey: after completing the field case they write that over $\mathcal{O}$ 'the conjecture remains open'.

Problem Statement

Let $p$ be an odd prime and $(\mathbb{K},\mathcal{O},\mathbb{F})$ a $p$ -modular system with $\mathbb{K}$ sufficiently large (containing the relevant roots of unity). For $n\geq 1$ , let $\tilde{\mathsf{S}}_n$ be a double cover of the symmetric group with central involution $z$ and $\tilde{\mathsf{A}}_n$ the corresponding double cover of the alternating group. Every spin block of $\mathcal{O}\tilde{\mathsf{S}}_n$ is labeled by a pair $(\rho,d)$ with $\rho$ a $\bar{p}$ -core, $d\in\mathbb{N}$ , and $n=|\rho|+dp$ ; for $0<d<p$ its defect group is abelian, isomorphic to $(C_p)^{\times d}$ . Prove or refute: for every odd prime $p$ , every weight $0<d<p$ , and every $\bar{p}$ -core $\rho$ (not only $d$ -Rouquier cores), the spin block $B^{\rho,d}$ of $\mathcal{O}\tilde{\mathsf{S}}_n(1-z)/2$ is derived equivalent to its Brauer correspondent over $\mathcal{O}$ , and likewise its even part $B^{\rho,d}_{\bar{0}}$ , a spin block of $\mathcal{O}\tilde{\mathsf{A}}_n$ , is derived equivalent to its own Brauer correspondent over $\mathcal{O}$ . This is Broué's abelian defect group conjecture over $\mathcal{O}$ for the spin blocks of the double covers: the required target is the derived equivalence over $\mathcal{O}$ stated above, whereas the expected strong form, matching what is known for RoCK blocks, is a splendid derived (splendid Rickard) equivalence implemented by bimodules with controlled vertices and sources. Establishing the splendid form is strictly stronger and also resolves the problem, and a counterexample refutes both forms; a proof of the derived equivalence over $\mathcal{O}$ without the splendid refinement therefore settles the problem as posed. A proof must cover the full parameter range: all odd primes $p$ , all $0<d<p$ , all $\bar{p}$ -cores $\rho$ , and both families $\tilde{\mathsf{S}}_n$ and $\tilde{\mathsf{A}}_n$ .

References

A. Kleshchev and M. Livesey, 'RoCK blocks for double covers of symmetric groups over a complete discrete valuation ring', Annals of Representation Theory 2 (2025) 85–171, doi:10.5802/art.22, arXiv:2309.14489, https://arxiv.org/abs/2309.14489
A. Kleshchev and M. Livesey, 'RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras', Memoirs of the American Mathematical Society 309 (2025), no. 1564, doi:10.1090/memo/1564, arXiv:2201.06870, https://arxiv.org/abs/2201.06870
Y. Du and X. Huang, 'On RoCK blocks of double covers of symmetric and alternating groups and the refined Broué conjecture', arXiv:2510.02147 (2025), https://arxiv.org/abs/2510.02147
J. Brundan and A. Kleshchev, 'Odd Grassmannian bimodules and derived equivalences for spin symmetric groups', Selecta Mathematica 32 (2026), no. 1 (published online 2025), doi:10.1007/s00029-025-01102-0, arXiv:2203.14149, https://arxiv.org/abs/2203.14149
M. Ebert, A. D. Lauda and L. Vera, 'Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications', arXiv:2203.14153 (2022), https://arxiv.org/abs/2203.14153
J. Chuang and R. Kessar, 'Symmetric groups, wreath products, Morita equivalences, and Broué's abelian defect group conjecture', Bulletin of the London Mathematical Society 34 (2002) 174–185, doi:10.1112/S0024609301008839, https://doi.org/10.1112/S0024609301008839
J. Chuang and R. 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
R. Kessar and M. Linckelmann, 'Descent of equivalences and character bijections', in: Geometric and topological aspects of the representation theory of finite groups, Springer Proceedings in Mathematics & Statistics 242 (2018) 181–212, doi:10.1007/978-3-319-94033-5_7, https://doi.org/10.1007/978-3-319-94033-5_7