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

Year 2024, Volume: 35 Issue: 35, 160 - 167, 09.01.2024

Çisil Karagüzel Deniz Yılmaz

https://doi.org/10.24330/ieja.1402947

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 $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that
\begin{align*}
\calT(B)\cong \Out_P(A)\rtimes \Out(P,\calF),
\end{align*}
where $\Out(P,\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\calF$ of $G$ on $P$ and $\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\times$.

Keywords

Block, fusion system, Picard group, dihedral defect group, generalised quaternion defect group

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