On Ext-Quivers of Blocks of Weight Two for Symmetric Groups

TL;DR

This paper develops an algorithm to analyze the Ext-quivers of weight 2 blocks of symmetric groups over fields of odd characteristic, establishing Morita equivalences based on Scopes classes for blocks with hook core partitions.

Contribution

The paper introduces a new algorithm relating Ext-quivers of weight 2 blocks forming a Scopes pair and explicitly determines quivers for blocks with hook cores.

Findings

Blocks with hook cores are Morita equivalent if and only if they are in the same Scopes class or conjugate.

The algorithm explicitly computes Ext-quivers for these blocks.

Morita equivalence classes are characterized by Scopes classes for weight 2 blocks.

Abstract

In this paper we investigate blocks of symmetric groups of weight 2 over fields of odd characteristic $p$. We develop an algorithm that relates the quivers of two such blocks forming a $(2:1)$ pair, as introduced by Scopes. We then apply the algorithm to blocks whose $p$-cores are hook partitions, in order to explicitly determine the quivers of these blocks. As a consequence we conclude that two $p$-blocks $B_1$ and $B_2$ of symmetric groups of weight $2$ whose $p$-cores are hook partitions are Morita equivalent if and only if $B_2$ or its conjugate block is in the same Scopes class as $B_1$.