Blocks with normal abelian defect and abelian p' inertial quotient

TL;DR

This paper studies blocks of group algebras with normal abelian defect groups and abelian inertial quotients, showing they are isomorphic to their Frobenius twist and providing explicit algebra descriptions.

Contribution

It proves isomorphism to Frobenius twist for such blocks and describes their basic algebra explicitly as a quantised semidirect product.

Findings

Blocks are isomorphic to their second Frobenius twist.

Explicit description of basic algebra as a quiver with relations.

Provides a quantised algebra structure of the block.

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 $G$ a finite group and $B$ a block of $\mathcal{O}G$ with normal abelian defect group and abelian $p'$ inertial quotient. 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 quantised version of the group algebra of the semidirect product $P\rtimes L$.