The strengthened Brou\'{e} abelian defect group conjecture for ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$

TL;DR

This paper proves that blocks of special linear and general linear groups over complete discrete valuation rings are splendidly Rickard equivalent to their Brauer correspondents, providing new evidence for a refined Broué conjecture.

Contribution

It establishes splendid Rickard equivalences for blocks of ${ m SL}(2,p^n)$ and ${ m GL}(2,p^n)$, supporting a refined Broué conjecture.

Findings

Blocks are splendidly Rickard equivalent to Brauer correspondents

Provides new evidence for Broué's abelian defect group conjecture

Supports a refined version of the conjecture by Kessar and Linckelmann

Abstract

We show that each $p$-block of ${\rm SL}(2,p^n)$ and ${\rm GL}(2,p^n)$ over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent, hence give new evidence for a refined version of Brou\'{e}'s abelian defect group conjecture proposed by Kessar and Linckelmann.