Let $G$ be finite group. I want to find simple and projective modules over the group algebra $A=\mathbb{C}[t][G]$. First observation is $A \cong \mathbb{C}[t]\otimes_{\mathbb{C}} \mathbb{C}[G] \cong \mathbb{C}[t] \otimes \oplus_{i=1}^r \operatorname{End}(S_i)$, where $S_i$ are simple $\mathbb{C}[G]$ modules.

Then we have the following simple modules $\mathbb{C}[t]/(t-\lambda) \otimes_\mathbb{C} S_i$. Is it true that there are no other simple modules over $A$?

Modules $\mathbb{C}[t] \otimes_\mathbb{C} S_i$ are projective because they are direct summands of $A$. Is it true that there are no other indecomposable projective modules over $A$?

Are there any books/papers where such representation theory of finite groups is discussed?


Since the algebra is a direct sum of matrix algebras over $\mathbb C[t]$, both irreducible and projective modules over this algebra are just modules over the summands. Moreover, modules over $\text{Mat}_n(R)$ correspond bijectively to modules over $R$ preserving simplicity and projectivity (Morita equivalence). Thus, we just need to classify simple and finite projective modules over $\mathbb C[t]$. And this is easy (any finite projective $\mathbb C[t]$-module is free).

Any textbook on graduate algebra (which talks about Morita equivalence) should be a good reference.


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