Representation theory of symmetric groups over Q-algebras

Question

Let $k$ be a commutative $\mathbb{Q}$-algebra, and let $\Sigma_n$ denote the symmetric group on $n$ letters. What is known about the classification of $k$-linear representations of $\Sigma_n$? (Of course, it will not be as concrete as the classification as in the case that $k$ is a field, but we may treat $k$-modules as already been classified and that these may be part of the classification data.) Can someone point me to relevant literature on this?

Answer 1

From the standard representation theory of $\Sigma_n$, it follows that the group algebra $\mathbb{Q}\Sigma_n$ is a product of matrix rings over $\mathbb{Q}$.

So $k\Sigma_n\cong\mathbb{Q}\Sigma_n\otimes_\mathbb{Q}k$ is a corresponding product of matrix rings over $k$, which is Morita equivalent to a product of copies of $k$.

Concretely, if $V_1,\dots,V_r$ are the irreducible $\mathbb{Q}\Sigma_n$-modules, then the $k\Sigma_n$-modules are of the form $$\bigoplus_{i=1}^r M_i\otimes_\mathbb{Q}V_i,$$ where $M_1,\dots,M_r$ are $k$-modules.