# Representations over $\mathbb{Q}_p$

I would like to understand representations over the $p$-adic field $\mathbb{Q}_p$ and find simple $\mathbb{Q}_p[G]$ modules for a finite group $G$. Is there some famous literature like for representations over $\mathbb{Z}$ from Curtis and Reiner? Especially I am looking for theorems with tells something about the number of simple $\mathbb{Q}_p[G]$ modules.

Thank you for hints

• Reiner's book "Maximal Orders" is pretty reasonable for Qp[G] and the descents to Zp[G]. Qp[G] is easier than Q[g] and harder than C[g]. Computing Schur indices over Qp is easier than over Q, is the only big change. – Jack Schmidt Apr 18 '13 at 14:56
• Thanks a lot for the reference. – user73227 Apr 18 '13 at 15:03

If $F$ is any field of char. zero, then $F[G]$ is a semisimple algebra, and general Wedderburn theory applies.

If we extend scalars to $\overline{F}$ (an alg. closure of $F$) then $\overline{F}[G]$ is a product of matrix rings over $\overline{F}$, and doesn't depend on which algebraically closed field you choose.

If $F$ is not algebraically closed, then $F[G]$ will be a product of matrix rings over division rings over $F$, and which ones appear depend on the details of the group $G$ and the field $F$ (basically, it depends on what algebraic integers do or don't already lie in $F$).

This should be explained in any reasonable text on group reps. (If you look up Schur index'' in the index of the book, this should take you to the section where non-algebraically closed fields are considered.)

In the case of $\mathbb Q_p$, there is a good theory of division rings over $p$-adic fields (part of local class field theory), and this will likely be a useful tool.

The analogy with $\mathbb Z[G]$ is not very good, because there the main point is that $\mathbb Z$ is not a field, so $\mathbb Z[G]$ is an order in a semisimple ring , rather than a semisimple ring itself.

• Are there also practical theorems like "the number of simple $\mathbb{Q}[G]$ modules is the number of different cyclic subgroups of $G$", for representations over the $p$-adic field? – user73227 Apr 18 '13 at 15:06
• @user73227: Dear user, Where are you getting the idea for such a possible "theorem"? If you take $G = S_3$, then the number of simple $\mathbb Q[G]$-modules (up to isomorphism) is three, while the number of different cyclic subgroups of $G$ is equal to $5$. So the statement you suggest is not true with $\mathbb Q$ coefficients, and nor will it be true with $\mathbb Q_p$-coefficients. Regards, – Matt E Apr 22 '13 at 1:58
• I found this theorem in the book "Representation theory of finite groups" from Curtis and Reiner in Corollar 39.5 – user73227 Apr 22 '13 at 11:31
• @user73227: Dear user, I think you have misstated that corollary; it is the number of conjugacy classes of cyclic subgroups. I would suggest that you learn the proof of the theorem, and you will then understand why it is not true for $\mathbb Q_p$: it is related to the fact that the $n$th cyclotomic polynomial is irreducible over $\mathbb Q$. So if the $|G|$th cyclotomic polynomial is irreducible over $\mathbb Q_p$ (e.g. if $p \not\mid |G|$ and $p$ is a primitive root mod $|G|$) then the same result will hold over $\mathbb Q_p$. Regards, – Matt E Apr 22 '13 at 17:34