Arbitrarily large $\mathcal{O}$-Morita Frobenius numbers

TL;DR

This paper constructs examples of finite group blocks with arbitrarily large Morita Frobenius numbers, suggesting these numbers are unbounded and addressing a question in modular representation theory.

Contribution

It provides the first known examples of blocks with large Morita Frobenius numbers, indicating these invariants can be arbitrarily large.

Findings

Constructed blocks with arbitrarily large $\

Suggests Morita Frobenius numbers are unbounded

Addresses a question of Benson and Kessar

Abstract

We construct blocks of finite groups with arbitrarily large $\mathcal{O}$-Morita Frobenius numbers. There are no known examples of two blocks defined over $\mathcal{O}$ that are not Morita equivalent but the corresponding blocks defined over $k$ are. Therefore, the above strongly suggests that Morita Frobenius numbers are also unbounded, which would answer a question of Benson and Kessar.