Splitting field for $\mathrm{GL}(2,p)$ - reference request

Question

It seems to me from a quick glance at several sources describing the complex and modular irreducible representations of $\mathrm{GL}(2,p)$ that any field $K$ containing a primitive $(p-1)$-root of unity is a splitting field for $\mathrm{GL}(2,p)$ (but maybe I need to read the details more carefully). Is this correct? And if so, what is a good source to cite for this besides folklore?

I flipped through a number of books, but they never bother to talk about field of definition for the irreducibles. But the constructions seem to always involve characters of $\mathbb F_p^\times$, which makes me suspect this is correct.

Edit. The original version of this question was incorrect, as pointed out in the comments. Here is the revised version.

1. Is any field of characteristic $0$ containing a primitive $(p^2-1)$-root of unity a splitting field for $GL(2,p)$? Note that $GL(2,p)$ has exponent $p(p^2-1)$, so I am trying to dump the factor $p$.
2. Is a any field of characteristic $p$ a splitting field for $\mathrm{GL}(2,p)$?

Answer 1

I realised that my previous version of this answer was irrelevant to the question (though what was said was accurate, as far as it went).

I think you can make the desired conclusion if you resort to modular representation theory, specifically, the theory of blocks with cyclic defect group (for this question we only need the cyclic defect theory with defect group of order $p$, which was done by R. Brauer himself). In a block with cyclic defect group, all decomposition numbers are $0$ or $1$. This means that every irreducible character occurs with multiplicity $1$ in some projective indecomposable module (over an appropriate local ring of characteristic zero).

Let $\chi$ be an irreducible character of a finite group $G$ which has order divisible by $p$ and not $p^{2}.$ Let $\theta$ be the character of a suitable projective indecomposable module as above, with $\langle \theta, \chi \rangle = 1.$

It is a theorem of W. F. Reynolds (a consequence of Brauer's characterization of characters), that $\theta$ is a $\mathbb{Z}$-linear combination of characters, each of which is induced from a Brauer elementary $p^{\prime}$-subgroups of $G$.

This means that $\chi$ occurs with multiplicity $1$ in the difference $\alpha - \beta$, where $\alpha$ and $\beta$ are characters of representations which are visibly respectively realizable over $\mathbb{K} = \mathbb{Q}[\omega]$, where $\omega$ is a primitive $m$-th root of unity and $m$ is the $p^{\prime}$-part of the exponent of $G$ ( in this case, just $\frac{e}{p},$ where $e$ is the exponent of $G$).

The Schur index $m_{\mathbb{K}}(\chi)$ divides both $\langle \alpha, \chi \rangle$ and $\langle \beta, \chi \rangle$ by the defining properties of the Schur index . Since $\langle \alpha - \beta, \chi \rangle = 1,$ we conclude that $m_{\mathbb{K}}(\chi) = 1.$

In the case of $G = {\rm GL}(2,p),$ this gives a positive answer to part $1$ of the question (every element of order divisible by $p$ has central $p^{\prime}$-part, and every element of order $p$ is conjugate to all its powers. Hence all character values on $p$-singular elements take values in the field $\mathbb{K}$ as above. and clearly all character values at $p$-regular elements take values in $\mathbb{K}$ (recall that $p$-regular means of order prime to $p$ and $p$-singular means of order divisible by $p$)).

Well, I suppose that strictly speaking, the answer as it stands does not now cover the irreducible characters of degree (divisible by) $p$,it only deals with the irreducible charcters in $p$-blocks of defect $1$. But for any irreducible character $\chi$ in a $p$-block of defect zero of any finite group $G$, the irreducible character $\chi$ is afforded by a projective $RG$-module, where $R$ is a suitable local ring of characteristic zero with residue field of characteristic $p$. Using the result of Reynolds, and arguing as above, we obtain $m_{\mathbb{K}}(\chi) = 1$, where $\mathbb{K}$ is the cyclotomic field extension of $\mathbb{Q}$ generated by a primitive $m$-th root of unity, with $m$ being the $p^{\prime}$-part of the exponent of $G$. In this case, $\chi$ vanishes on elements of order divisible by $p$, so that all character values taken by $\chi$ are in $\mathbb{K}.$

Answer 2

(too long for a comment.)

1. The character table of GL(2,p) in characteristic 0 can be found for example the book "A Journey through representation theory" by Gruson and Serganova on page 147.

2. seems to be true, but I do not know a reference (maybe books by Gordon James?). Here is a way to test it for a given prime with MAGMA (you can input it in http://magma.maths.usyd.edu.au/calc/ ). The code also works for any other finite group to get a minimal splitting field over prime fields.

   p:=7; G:=GeneralLinearGroup(2, GF(p)); SIMS := AbsolutelyIrreducibleModules(G,GF(p)); temp_field_sizes:=[]; for i in SIMS do Append(~temp_field_sizes,#BaseRing(i)); end for; MaX := Maximum(temp_field_sizes); MaX;

when the result is p again, then the prime field is a splitting field. Magma confirms that the prime field is a splitting field for all primes <20.