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.

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)