An algorithmic approach to perverse derived equivalences: Brou\'e's Conjecture for $\Omega^{+}_8(2)$

TL;DR

This paper develops algorithms to verify Broué's conjecture for certain blocks of finite groups, successfully applying them to prove the conjecture for the principal 5-block of (2).

Contribution

It introduces two algorithms implementing a computational approach to Broue9's conjecture for blocks with abelian defect groups, and applies them to a specific complex group case.

Findings

Broue9's conjecture verified for (2)

Algorithms successfully construct stable and perverse derived equivalences

Provides computational tools for future verification of conjectures in modular representation theory.

Abstract

Following Craven and Rouquier's computational method to tackle Brou\'e's abelian defect group conjecture, we present two algorithms implementing that procedure in the case of principal blocks of defect $D \cong C_{\ell} \times C_{\ell}$ for a prime $\ell$; the first describes a stable equivalence between $B_0(G)$ and $B_0(N_G(D))$, and the second tries to lift a such stable equivalence to a perverse derived equivalence. As an application, we show that Brou\'e's conjecture holds for the principal $5$-block of the simple group $\Omega^{+}_8(2)$.