Block extensions, local categories, and basic Morita equivalences

TL;DR

This paper studies invariants of block extensions in modular representation theory, introducing local categories, group extensions, and cohomology classes, and proves their invariance under certain Morita equivalences, also providing alternative proofs of existing results.

Contribution

It introduces new invariants for block extensions and proves their invariance under $G/H$-graded basic Morita equivalences, extending and simplifying previous results.

Findings

Invariants include extended local categories, group extensions, and cohomology classes.

These invariants are preserved under $G/H$-graded basic Morita equivalences.

Provides alternative proofs for known results on nilpotent and inertial blocks.

Abstract

Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system with $k$ algebraically closed, let $b$ be a block of the normal subgroup $H$ of $G$ having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$, and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta)/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha]\in H^2(N_G(Q_\delta)/QC_H(Q),k^\times)$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of K\"ulshammer and Puig (1990), Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p'$-extensions of inertial blocks.