The Kessar–Linckelmann refinement of Broué's abelian defect group conjecture: splendid Rickard equivalences over arbitrary complete discrete valuation rings

For a fixed prime $p$, let $\mathcal{O}$ be a complete discrete valuation ring of characteristic $0$ with residue field $k$ of characteristic $p$ — not assumed to be a splitting field — and let $G$ be an arbitrary finite group. The Kessar–Linckelmann refinement of Broué's abelian defect group conjecture predicts that whenever a block algebra $\mathcal{O}Gb$ has an abelian defect group $D$, it is splendidly Rickard equivalent to its Brauer correspondent $\mathcal{O}N_G(D)c$: there is a bounded two-sided complex of $p$-permutation bimodules, with vertices on the diagonal copy of $D$, inducing the equivalence. The conjecture is open in general; work spanning 2008–2026 has verified it only for restricted families such as symmetric and alternating groups, unipotent blocks (a distinguished family of blocks of finite general linear groups) of $GL_n(q)$ in non-defining characteristic (the block prime does not divide $q$), the groups $SL(2,p)$ and $GL(2,p)$, and $p$-nilpotent groups (groups with a normal $p$-complement, that is, a normal subgroup of $p$-power index and order prime to $p$) with abelian Sylow $p$-subgroup.

Background

In the modular representation theory of finite groups one works over a $p$-modular system: a complete discrete valuation ring $\mathcal{O}$ of characteristic $0$ with residue field $k$ of characteristic $p$ (e.g. the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_p$). The group algebra $\mathcal{O}G$ decomposes into blocks: $\mathcal{O}G=\bigoplus_b \mathcal{O}Gb$ for central primitive idempotents $b$, and each block algebra $\mathcal{O}Gb$ carries an invariant called its defect group, a $p$-subgroup $D$ of $G$ defined up to conjugacy, which measures how far the block is from being semisimple. Brauer's first main theorem states that conjugacy classes of blocks of $\mathcal{O}G$ with defect group $D$ correspond bijectively to conjugacy classes of blocks of the normalizer $N_G(D)$ with defect group $D$; the block $c$ of $\mathcal{O}N_G(D)$ corresponding to $b$ is called its Brauer correspondent, so the 'local' algebra $\mathcal{O}N_G(D)c$ is canonically paired with the 'global' algebra $\mathcal{O}Gb$. Broué's abelian defect group conjecture (circa 1990) asserts that when $D$ is abelian, $\mathcal{O}Gb$ and $\mathcal{O}N_G(D)c$ are derived equivalent (their bounded derived categories of modules are equivalent), over a splitting coefficient system — a residue field over which all relevant group algebras split. Rickard strengthened the expected conclusion to a splendid Rickard equivalence: a bounded complex $X$ of $(\mathcal{O}Gb,\mathcal{O}N_G(D)c)$-bimodules, projective on both sides, with $X\otimes X^{*}$ homotopy equivalent to $\mathcal{O}Gb$ and $X^{*}\otimes X$ homotopy equivalent to $\mathcal{O}N_G(D)c$, whose terms are $p$-permutation modules (direct summands of permutation modules) as modules for the direct product $G\times N_G(D)$, with indecomposable summands having vertices contained in the diagonal subgroup $\Delta D=\{(u,u)\mid u\in D\}$ (the vertex of an indecomposable module is a minimal subgroup of the acting group from which the module can be obtained as a direct summand of an induced module). A splendid Rickard equivalence is in particular a derived equivalence, but its $p$-permutation structure carries local information that plain derived equivalences do not. Kessar and Linckelmann (Springer Proc. Math. Stat. 242, 2018) proposed the refined version studied here: drop the splitting hypothesis, so $\mathcal{O}$ is an arbitrary complete discrete valuation ring with residue field of characteristic $p$, and ask for a splendid Rickard equivalence for every block with an abelian defect group of every finite group. The refinement is not cosmetic. Categorical equivalences between blocks induce character bijections compatible with Galois groups of field extensions, and Boltje's Broué invariant — a canonical character-theoretic invariant attached to any $p$-permutation equivalence — shows that descending a splendid Rickard equivalence to $\mathbb{Z}_p$ implies the Isaacs–Navarro, Navarro, and Turull refinements of the Alperin–McKay conjecture for the block, so the refined conjecture strengthens the local–global counting conjectures of character theory. (The Alperin–McKay conjecture itself predicts that a block and its Brauer correspondent have equal numbers of height-zero irreducible characters, that is, irreducible characters whose degrees take the minimal $p$-power divisibility permitted within the block.) Known cases include blocks with cyclic defect groups (Rouquier, with descent to non-splitting rings by Kessar–Linckelmann), blocks of symmetric groups with non-trivial abelian defect (Chuang–Rouquier, via $\mathfrak{sl}_2$-categorification, a lift of the action of the Lie algebra $\mathfrak{sl}_2$ on weight spaces to exact functors on module categories), alternating groups (Marcus over splitting rings; Huang over arbitrary complete discrete valuation rings), and several further families documented below. Over splitting coefficient systems the conjecture specializes to Broué's original conjecture, which remains open in general after more than three decades.

Problem Statement

Prove or disprove the Kessar–Linckelmann refinement of Broué's abelian defect group conjecture: for every prime $p$, every complete discrete valuation ring $\mathcal{O}$ of characteristic $0$ with residue field of characteristic $p$ (the residue field is not assumed to be a splitting field for the groups involved), every finite group $G$, and every block idempotent $b$ of $\mathcal{O}G$ whose defect group $D$ is abelian, the block algebra $\mathcal{O}Gb$ is splendidly Rickard equivalent to its Brauer correspondent $\mathcal{O}N_G(D)c$. Concretely: there must exist a bounded complex $X$ of $(\mathcal{O}Gb,\mathcal{O}N_G(D)c)$-bimodules, projective as one-sided modules, such that $X\otimes_{\mathcal{O}N_G(D)c}X^{*}$ and $\mathcal{O}Gb$ are homotopy equivalent as complexes of $(\mathcal{O}Gb,\mathcal{O}Gb)$-bimodules, $X^{*}\otimes_{\mathcal{O}Gb}X$ and $\mathcal{O}N_G(D)c$ are homotopy equivalent as complexes of $(\mathcal{O}N_G(D)c,\mathcal{O}N_G(D)c)$-bimodules, and each term of $X$, as an $\mathcal{O}(G\times N_G(D))$-module, is a $p$-permutation module whose indecomposable summands have vertices contained in the diagonal subgroup $\Delta D=\{(u,u)\mid u\in D\}$. A disproof would consist of a specific finite group, prime, complete discrete valuation ring, and block with abelian defect group for which no such complex exists, together with a proof of nonexistence.

References

1. Radha Kessar, Markus Linckelmann, “Descent of Equivalences and Character Bijections”, Springer Proceedings in Mathematics & Statistics 242 (2018) 181–212, DOI 10.1007/978-3-319-94033-5_7, arXiv:1705.07227, https://doi.org/10.1007/978-3-319-94033-5_7
2. Xin Huang, “Descent of splendid Rickard equivalences in alternating groups”, Journal of Algebra 658 (2024) 277–293, DOI 10.1016/j.jalgebra.2024.05.048, arXiv:2111.10922, https://doi.org/10.1016/j.jalgebra.2024.05.048
3. Andrei Marcus, “Broué’s abelian defect group conjecture for alternating groups”, Proceedings of the American Mathematical Society 132 (2004) 7–14, DOI 10.1090/s0002-9939-03-07214-9, https://doi.org/10.1090/s0002-9939-03-07214-9
4. Joseph Chuang, Raphaël Rouquier, “Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification”, Annals of Mathematics 167 (2008) 245–298, DOI 10.4007/annals.2008.167.245, https://doi.org/10.4007/annals.2008.167.245
5. Raphaël Rouquier, “The derived category of blocks with cyclic defect groups”, in Derived Equivalences for Group Rings, Lecture Notes in Mathematics 1685, Springer (1998) 199–220, DOI 10.1007/bfb0096376, https://doi.org/10.1007/bfb0096376
6. Robert Boltje, “The Broué invariant of a $p$-permutation equivalence”, Annals of Representation Theory 1 (2024) 517–537, DOI 10.5802/art.17, arXiv:2007.13936, https://doi.org/10.5802/art.17
7. Xin Huang, Pengcheng Li, Jiping Zhang, “The strengthened Broué abelian defect group conjecture for $SL(2,p)$ and $GL(2,p)$”, Journal of Algebra 633 (2023) 114–137, DOI 10.1016/j.jalgebra.2023.05.049, https://doi.org/10.1016/j.jalgebra.2023.05.049
8. Xin Huang, Pengcheng Li, Jiping Zhang, “Descent of splendid Rickard equivalences in $GL_n(q)$”, Journal of Pure and Applied Algebra 228 (2024) 107711, DOI 10.1016/j.jpaa.2024.107711, arXiv:2302.12129, https://doi.org/10.1016/j.jpaa.2024.107711
9. Sam K. Miller, “Galois descent of splendid Rickard equivalences for blocks of $p$-nilpotent groups”, Proceedings of the American Mathematical Society (2025), DOI 10.1090/proc/17230, arXiv:2405.16061, https://doi.org/10.1090/proc/17230
10. Jadyn V. Breland, Sam K. Miller, “Brauer pairs for splendid Rickard equivalences”, Journal of Algebra 691 (2026) 694–729, DOI 10.1016/j.jalgebra.2025.11.029, arXiv:2312.10258, https://doi.org/10.1016/j.jalgebra.2025.11.029