# Descent of splendid Rickard equivalences in ${\rm GL}_n(q)$

## TL;DR

This paper proves a refined version of Broué's abelian defect group conjecture for unipotent blocks of GL_n(q) and provides a condition for general blocks, illustrating its limitations with an example.

## Contribution

It establishes a refined conjecture for unipotent blocks of GL_n(q) and offers a sufficient condition for other blocks, highlighting its limitations with counterexamples.

## Key findings

- Refined Broué conjecture holds for unipotent blocks of GL_n(q).
- A sufficient condition is identified for general blocks to satisfy the conjecture.
- Counterexample shows the condition does not always hold.

## Abstract

Let $n$ be a positive integer and $q$ a prime power. We prove that a refined version of Brou\'{e}'s abelian defect group conjecture holds for unipotent $\ell$-blocks of ${\rm GL}_n(q)$, where $\ell\nmid q$. We also give a sufficient condition on general $\ell$-blocks of ${\rm GL}_n(q)$ to satisfy the refined abelian defect group conjecture. We explain by an example that this sufficient condition does not hold in general.