# Representation theory of symmetric groups over rings

This is a continuation of my previous question. Let $R$ be a commutative ring in which $n!$ is invertible. Does it follow that every $R \Sigma_n$-module decomposes as a direct sum of modules of the form $V_i \otimes_R M_i$, where $M_i$ is some $R$-module and $V_i$ is some $R \Sigma_n$-module which is a direct summand of $R \Sigma_n$?

Notice that, even if we had $R \Sigma_n \cong \prod_i M_{n_i}(R)$ (is this true?), and $V_i$ is chosen to be $W_i \otimes_{\mathbb{Z}} R$, where $W_i$ is some integral model for a irreducible complex representation $W_i \otimes_{\mathbb{Z}} \mathbb{C}$ of $\Sigma_n$, the usual projection $\mathbb{C} \Sigma_n \to W_i \otimes_{\mathbb{Z}} \mathbb{C}$ does not seem to be definable over $\mathbb{Z}[1/n!]$, and therefore it is not clear to me if $V_i$ is a direct summand of $R \Sigma_n$.

• I get a little nervous about finiteness conditions on $R$. Based on your comments to the linked question, are you really interested in $R=\mathbb{Z}[1/n!]$? – David Hill Mar 31 '17 at 16:45
• @DavidHill: This is an example, yes. In general, $R$ is a $\mathbb{Z}[1/n!]$-algebra. I am not sure which finiteness conditions you want to impose. – HeinrichD Mar 31 '17 at 16:46
• If you prove that $R \Sigma_n$ is a product of matrix algebras over $R$, then its representation theory is "nice", as you're trying to describe - although you don't want $V_i$ to be irreducible in general. Actually, the right choice for $V_i$ for each matrix algebra is the space of column vectors with entries in $R$ that it acts on - this expresses the Morita equivalence of $R$ with a matrix algebra over $R$. – Dustan Levenstein Mar 31 '17 at 16:57
• The name for the "integral model" that you reference is Specht module - they can be defined over the integers. The question is precisely whether or not $R \Sigma_n$ is the product of $R$-endomorphism algebras of the appropriate Specht modules; this would enable one to define the projection maps that you're referencing. – Dustan Levenstein Mar 31 '17 at 18:05
• The notes by Garsia I've referenced in my answer to math.stackexchange.com/a/2213581 should (once shaken well) provide an isomorphism from $R \Sigma_n$ to a direct product of matrix algebras over $\mathbb{Z}\left[1/n!\right]$. – darij grinberg Apr 1 '17 at 20:28