Vertices of modules and decomposition numbers of $C_2 \wr S_n$

TL;DR

This paper investigates the structure of modules in the wreath product of a cyclic group of order 2 with the symmetric group, classifies their vertices, and describes parts of their decomposition matrix in odd prime characteristic.

Contribution

It provides a classification of module vertices and describes specific columns of the decomposition matrix for the wreath product $C_2 times S_n$ in odd prime characteristic.

Findings

Vertices of modules are classified for the wreath product $C_2 times S_n$.

Certain columns of the decomposition matrix are explicitly described.

The structure of modules with involution model characters is elucidated.

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 $C_2 \wr S_n.$