Reference request: Representation theory over fields of characteristic zero

Question

Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook on representation theory which works with fields of characteristic zero directly from the beginning. (In case there are many such textbooks, I would prefer "modern" ones which emphasize the theory of non-commutative algebras; it should really be the opposite of Fulton-Harris.) In particular, it should mention the classification of irreducible representations of symmetric groups over $\mathbb{Q}$, and there should be more than just a side remark that those over $\mathbb{C}$ descend to $\mathbb{Q}$.

Answer 1

The book "The representation theory of the symmetric group" (Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley, Reading, Mass., 1981), by Gordon James and Adalbert Kerber, develops the characteristic zero representation theory of symmetric groups over $\mathbb{Q}$. In particular it has a theorem (Theorem 2.1.12) explicitly stating that every field is a splitting field for $S_n$.

Answer 2

Here is a list of references that should be somewhat useful. Unfortunately, I am not aware of a single reference that takes you reasonably far all by itself -- and I'd much like to!

• Martin Lorenz, A tour of representation theory (a draft, but most of the text is fully fleshed out already) does lots of general representation theory in high generality. Unfortunately, as it comes to symmetric groups, it relies on algebraic closedness due to its use of the Okounkov-Vershik approach. Still, you should be able to extract a lot of good material out of it. (I am afraid that this is an innate downside of the Okounkov-Vershik approach -- it uses "diagonalizability through positivity" results -- but I'd be happy to learn otherwise!)

• Mark Wildon, Representation theory of the symmetric group is a really nice combinatorial introduction into the theory. At one point it uses a "Theorem (Brauer)" that is probably not easy to prove elementarily, but by that point a lot has already been proven.

• Adriano Garsia, Young seminormal representation: Murphy elements and content evaluation is an enlightening resource on the seminormal form, which is a very elementary (if rather intransparent and computational) approach to the representations of the symmetric group. Unfortunately, these notes are rough and have errors (e.g., the proof of Proposition 2.3 is wrong; instead, the proposition can be proven by computing $e_{11}^\lambda$ in two different ways: once using (2.26), resulting in $1 / h_\lambda$; once again as the trace of $e_{11}^\lambda$ acting by left multiplication on the $\left(e_{ij}^\mu\right)$ basis of $\mathcal{A}\left(S_n\right)$, resulting in $n_\lambda/n!$). They also omit some proofs (e.g., the "cute argument" referenced in the proof of Theorem 1.2 says that if $f$ is a nilpotent element of a group algebra, then the trace of $f$ must be a multiple of the coefficient of the identity in $f$, but on the other hand, this trace must be $0$ since $f$ is nilpotent).

• Possibly, Chapter 7 of William Fulton, Young Tableaux, LMS Student Texts 35, Cambridge University Press 1997 also doesn't rely on the algebraic closure of the base field. (I don't know for sure.)

Likely you'll need to combine different sources.