# Does it make sense that “Representations of groups over finite ring” ?

I am an undergrad student who wants to know about the representation theory over

arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular

representation theory.)

In particular, I would like to study the representation theory of modular Heisenberg groups

(entries in a finite field)over finite rings. I will start with the homomorphisms from

this Heisenberg groups to the ring of integers and the finite ring of integers.

Actually, I mean that does it make sense "over finite ring".

If it does, I would like to know good references and brief explanations.

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Dear Lee, There is, of course, much to be said about the representation theory of groups over finite rings. Indeed, there's much more than can be said in a MathOverflow post, and as such, your question is a bit more open-ended than would be best for this site. Please read mathoverflow.net/howtoask and consider revising this question to be more specific: what do you already know, and what in particular would you like to know? – Theo Johnson-Freyd Jun 28 '11 at 14:07
I believe Lee was wondering whether there was some good reason we shouldn't do representations over finite rings (there isn't), and if not, he would like a good reference to go read about them. I don't have one, but someone should post one. – Richard Rast Jun 28 '11 at 16:54
@David: That's not necessarily true. In modular representation theory $\mathcal{O}[G]$ (where $\mathcal{O}$ is a discrete valuation) and its modules are useful and important objects linking the representation theory over the quotient field (which is usually chosen to have characteristic 0) and the residue field (which is usually chosen to have positive characteristic) of $\mathcal{O}$. – Johannes Hahn Jun 28 '11 at 20:59
@Richard, David and Dan: I was meant to say maps from groups to matrices over a finite ring. I'll edit my question. – Lee Jun 29 '11 at 0:32
Your revised question amounts to studying maps of the form $\mathbb{F}_q[H] \to M_n(R)$, where $H$ is the Heisenberg group, and $R$ is your finite ring. This certainly makes sense, and Curtis and Reiner's book might be a decent reference. – S. Carnahan Jun 29 '11 at 3:17

EDIT: Everything below the line is from before the OP changed his question. It was attempting to study linear maps $\rho: R \rightarrow GL(V)$ for $R$ a finite ring and $V$ a vector space. Now that the question has been cleared up, we see that the OP desired to study $\rho: G \rightarrow GL(R)$ for $G$ a group and $R$ a ring. Here's my attempt to see this question resolved, following a recommended procedure in meta by linking to the references given in the comments above and making my answer CW so I don't get rep for this.

S. Carnahan mentioned Curtis and Reiner's text Representation theory of finite groups and associative algebra. A few moments of browsing shows me that page 30 has the first definition of representation and one can see from there how to develop the theory the OP sought.

Dan Ramras recommends Dave Benson's book Representations and Cohomology: Basic representation theory of finite groups and associative algebras. Note that this book comes in two parts, and the OP almost certainly wants the first part since the second is highly topological and focuses on group cohomology.

@Lee If these answer your question you can accept this answer by clicking the check mark next to this box. That way it won't appear on the front page again unless someone else answers or edits.

I did some googling and found a few references. First, this article reduces a problem the authors are interested in to a well known problem in the representation theory of finite rings. At the end are some references to that problem and those should include the necessary background you need to get into this subject.

Another reference is this MO post, which includes as many freely available texts on representation theory of groups as could be found. Perhaps one of them discusses or mentions representation theory of rings as well.

In case none of those references pan out, here is perhaps a place to start your thinking. A representation of a group is $\rho: G \rightarrow GL(\mathbb{C})$ and is equivalent to a module over the group ring $\mathbb{C}[G]$. A representation of a ring would need to be $\rho: R\rightarrow GL(\mathbb{C})$ and should be equivalent to a module $M$ over a more general ring. We can form det $\circ \rho$: from the units of $R$ to $\mathbb{C}$, and this restricts the form $M$ can take.

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Thanks. Curtis and Reiner would be very helpful. Also, I found other two books of Curtis and Reiner(Methods of representation theory vol 1&2)are extreamly wonderful. – Lee Jun 29 '11 at 9:59