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

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 $D \cong C_{\ell} \times C_{\ell}$foraprime$\ell$;thefirstdescribesastableequivalencebetween$B_0(G)$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 ${\mathrm{\Omega }}_8^{+}(2)$.