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

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 $B_1$.