Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.

Question

This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation of the following:

Let $S_n$ be the symmetric group of $n$ symbols and let $F=\mathbb{F}_q$ be a finite field of $q$ elements and characteristic $p>n$. Then $F$ is a splitting field for $S_n$ (that is to say every irreducible $FS_n$-module is absolutely irreducible). Furthermore, every irreducible representation of $S_n$ over $F$ is also irreducible over $\mathbb{C}$.

This seems almost tautological to me, but it would be nice to have confirmation just to make sure I haven't misunderstood thinks spectacularly.

As a further (potentially stupid) question, are these irreducible representations over $F$ the 'same' as those over $\mathbb{C}$, albeit with reduction modulo $p$. Do they have the same degree for instance? Again from the definitions, this would appear to me to be an emphatic yes, but again it would be nice to have confirmation.

Thank you.

Answer 1

One can show that for every $\def\bb{\mathbb}\bb CS_n$-module $V$ there is a $\bb ZS_n$-module $V'$ which is free as an abelian group and of rank equal to the dimension of $V$ as a vectr space, such that $V'\otimes_{\bb Z}\bb C\cong V$ as $\bb CS_n$-modules. This is a rather special property of $S_n$.

Now, if we start with $V$, construct $V'$ as above, then we can also construct $V''=V'\otimes_{\bb Z}\bb F_p$, and this is a representation of $\bb F_pS_n$. The passage from $V$ to $V''$ involves a choice, that of $V'$. When one understands what it entails, one can wonder if the mapping $V \leadsto V''$ gives a bijection between irreps of $\bb CS_n$ an irreps of $\bb F_pS_n$. It does, as long as $p>n$.

Other finite groups which have this property that its complex irreps are defined over $\bb Z$ are the Weyl groups (of which $S_n$ is an example) and surely others. For a general group $G$, a complex representation $V$ is always defined over the ring $\mathcal O$ of integers of some splitting field of $G$ (one can show that there is a splitting field which is cyclotomic) and then one can "reduce" $V$ modulo prime ideals of $\mathcal O$. The theory is extremely rich, complicated and deep. You can find this explain in the classic book by Curtis and Reiner, for example. The litle book of Serre has lots of information on the problem of rationality of representations, that is, of trying to restrict the field over which a representation is defined, but here we want considerably more; we want to restrict to a ring whose field of quotients is a field of definition of the representation. Already for cyclic groups this is a non-obvious problem.