Fusions and Clifford extensions

TL;DR

This paper introduces a new fusion concept called fusions of local pointed groups in block extensions, demonstrating their invariance under group graded basic Morita equivalences, advancing modular representation theory.

Contribution

It defines fusions in the context of block extensions and proves their invariance under specific Morita equivalences, providing new tools for studying block invariants.

Findings

fusions generalize classical fusion concepts.

Clifford extensions associated to pointed groups are invariant under group graded Morita equivalences.

The results deepen understanding of block invariants in modular representation theory.

Abstract

We introduce $\bar G$-fusions of local pointed groups on a block extension $A=b\mathcal{O}G$, where $H$ is a normal subgroup of the finite group $G$, $\bar G=G/H$, and $b$ is a $G$-invariant block of $\mathcal{O}H$. We show that certain Clifford extensions associated to these pointed groups are invariant under group graded basic Morita equivalences.