On a question of Rickard on tensor product of stably equivalent algebras

TL;DR

This paper investigates the stable equivalence of Morita type between certain blocks of group algebras over algebraically closed fields, showing that their centers are not isomorphic in specific cases, thus answering Rickard's question negatively.

Contribution

It demonstrates that the principal block of $ar{ extbf{F}}_pPSU(3,p^r)$ is not stably equivalent of Morita type to its Brauer correspondent in certain cases, providing a counterexample to a conjecture.

Findings

Centers of the blocks are not isomorphic in specified cases.

Stable equivalence of Morita type does not hold after tensoring with certain polynomial algebras.

Provides a negative answer to Rickard's question about tensor products of stably equivalent algebras.

Abstract

Let $\overline\F\_p$ be the algebraic closure of the prime field of characteristic $p$. After observing that the principal block $B$ of $\overline\F\_pPSU(3,p^r)$ is stably equivalent of Morita type to its Brauer correspondent $b$, we show however that the centre of $B$ is not isomorphic as an algebra to the centre of $b$ in the cases $p^r\in\{3,4,5,8\}$. As a consequence, the algebra $B\otimes\_{\overline{\F}\_p}\overline \F\_p[X]/X^p$ is not stably equivalent of Morita type to $b\otimes\_{\overline\F\_p}\overline\F\_p[X]/X^p$ in these cases. This yields a negative answer to a question of Rickard.