3
$\begingroup$

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?

$\endgroup$
-1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.