On RoCK blocks of double covers of symmetric and alternating groups and the refined Broué conjecture

TL;DR

This paper advances the refined Broué abelian defect group conjecture for RoCK blocks of spin double covers of symmetric and alternating groups by showing that Kleshchev–Livesey's Morita superequivalences and splendid Rickard equivalences descend from large coefficient rings to the ring of $p$-adic integers $ ext{Z}_p$. Central to the approach are RoCK blocks labelled by $( ho,d)$, twisted wreath superproducts, and endopermutation sources, enabling a precise tensor-structural description: $B^{ heta, ho,d}_{R_p}$ is Morita superequivalent to $B^{ heta, ho,0}_{R_p} ensor_{R_p}(B^{ heta, abla,1}_{R_p} times_s ext{T}^ heta_{d,R_p})$ with endopermutation source $U^{R_p}$, and splendid Rickard equivalences to Brauer correspondents descend as well. Over $ ext{F}_p$ these correspondences are established explicitly, and lifting arguments yield the $ ext{Z}_p$-level splendid Rickard equivalences, validating the refined conjecture for RoCK spin blocks. Consequently, refinements of Navarro–Turull–Boltje-type conjectures hold for these blocks, and the methodology provides a robust descent framework for non-splitting settings in modular representation theory. The results solidify the connection between local RoCK data and global spin block structure in the $p$-adic context, with potential further applications to related block correspondences.

Abstract

Recently, Kleshchev and Livesey proved the existence of RoCK $p$-blocks for double covers of symmetric and alternating groups over large enough coefficient rings. They proved that these RoCK blocks of double covers are Morita equivalent to standard ``local" blocks via bimodules with endopermutation source. Based on this, Kleshchev and Livesey proved that RoCK blocks are splendidly Rickard equivalent to their Brauer correspondents. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué's abelian defect group conjecture for symmetric groups. In this paper we show that the Morita and splendid Rickard equivalences constructed by Kleshchev and Livesey descend to the ring $\mathbb{Z}_p$ of $p$-adic integers, hence prove Kessar and Linckelmann's refinement of Broué's abelian defect group conjecture for these RoCK blocks.

Theorems & Definitions (64)

• Conjecture 1.1: The refined Broué conjecture Kessar_Linckelmann
• Theorem 1.2: KL22
• Theorem 1.3: Theorem \ref{['theo: Brauer corr over Fp']}
• Theorem 1.4: KL22
• Theorem 1.5: Theorem \ref{['theorem:Brauer corr over Fp splendid']}
• Theorem 1.6: KL22, KL25
• Theorem 1.7: KL25
• Theorem 1.8: Theorem \ref{['theorem: Proof of Theorem main1']}
• Theorem 1.9: Theorem \ref{['theo: proof of main2']}
• Corollary 1.10