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Splendid equivalence formulation of Broué’s conjecture

Establish the existence of a splendid equivalence between the principal blocks of OG and OH for finite groups G and H with a common Sylow p-subgroup and the same p-local structure; concretely, construct a split-endomorphism two-sided tilting complex of OG.e–OH.f-bimodules with terms that are direct summands of relatively ΔP-projective permutation modules, inducing an equivalence between the bounded derived categories of the principal blocks.

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Background

Rickard introduced splendid equivalences to structurally explain families of perfect isometries (isotypies) predicted by Broué. Under suitable local structure hypotheses, a splendid equivalence between principal blocks yields compatible derived equivalences at all p-local levels, matching character-theoretic data.

The paper states this strengthened, ‘splendid’ formulation of Broué’s conjecture and notes that, despite many known cases, it remains open in general.

References

Following Theorem~\ref{thm:splendidinducesisotpy}, Brou\ e's abelian defect group conjecture can be restated as:

\begin{conjecture} \label{conj:Brouesplendid} Let $G$ and $H$ be two finite groups with a common Sylow $p$-subgroup and same $p$-local structure. Then, there is a splendid equivalence between the principal blocks of $ \operatorname{O} G$ and $ \operatorname{O} H$. \end{conjecture}

Just as Brou\ e's original abelian defect group conjecture (Conjecture \ref{conj:Brouederived}), this strenghtened version of the conjecture (Conjecture \ref{conj:Brouesplendid}) is still open in general.

— Rickard's Derived Morita Theory: Review and Outlook (2509.06369 - Jasso et al., 8 Sep 2025) in Conjecture, Subsection 1.4 (Modular representation theory and splendid equivalences)