BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

Abstract

Abstract Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.