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

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    $\begingroup$ 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. $\endgroup$ – Jack Schmidt Apr 18 '13 at 14:56
  • $\begingroup$ Thanks a lot for the reference. $\endgroup$ – 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.

  • $\begingroup$ 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? $\endgroup$ – user73227 Apr 18 '13 at 15:06
  • $\begingroup$ @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, $\endgroup$ – Matt E Apr 22 '13 at 1:58
  • $\begingroup$ I found this theorem in the book "Representation theory of finite groups" from Curtis and Reiner in Corollar 39.5 $\endgroup$ – user73227 Apr 22 '13 at 11:31
  • $\begingroup$ @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, $\endgroup$ – Matt E Apr 22 '13 at 17:34

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