The group of splendid Morita equivalences of principal $2$-blocks with dihedral and generalised quaternion defect groups

Abstract

Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\mathcal{T}(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \begin{align*} \mathcal{T}(B)\cong \mathrm{Out}_P(A)\rtimes \mathrm{Out}(P,\mathcal{F})\,, \end{align*} where $\mathrm{Out}(P,\mathcal{F})$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\mathcal{F}$ of $G$ on $P$ and ${\mathrm{Out}}_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P{)}^{\times }$.