Vertices of Modules and Decomposition Numbers of C2 ≀ Sn

Abstract

Given $n \in \mathbf{N},$ consider the imprimitive wreath product $C_2 \wr S_n.$ We study the structure of modules whose ordinary characters form an involution model of $FC_2 \wr S_n,$ where $F$ is a field of odd prime characteristic. We classify the vertices of these modules in this case. We then use this classification of the vertices to describe certain columns of the decomposition matrix of $C_2 \wr S_n.$