An inequality on representations and subgroups of general linear groups over finite field

Question

Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H_1$ and $H_2$ be two subgroups of $GL_n(\mathbb F_q)$ that are $l$-groups.

If for all characteristic $0$ representations $V$ of $GL_n(\mathbb F_q)$, we have

$$\dim V^{H_1} \leq \dim V^{H_2}$$

does it follow that:

$$\dim \left(\mathbb F_q^n\right)^{H_1} \leq \dim \left(\mathbb F_q^n\right)^{H_2}$$?

I'm thinking about a problem that would be significantly simplified if this were true, although I suspect it is false. I would do a large finite search for counterexamples before asking about this question, but I don't know a way to efficiently search for counterexamples.

Answer 1

The following is a series of counterexamples: Let $q$ be a prime power with $q\equiv1\pmod{8}$, and pick $\omega\in\mathbb F_q^\star$ with order $8$. Set $G=\text{GL}_2(\mathbb F_q)$, and let $H_1$ and $H_2$ be the cyclic subgroups generated by $\begin{pmatrix}\omega & 0\\0 & 1\end{pmatrix}$ and $\begin{pmatrix}\omega^2 & 0\\0 & \omega^4\end{pmatrix}$, respectively. Of course, the second inequality does not hold.

The characteristic $0$ inequalities can be verified by the explicit form of the character table of $G$, as given for instance in Theorem 28.5 in this book by James and Liebeck (see also this write up for another readable account).

Note that if $m$ is a multiple of the order of $g\in G$, then the dimension of the fixed space of $g$ in the representation afforded by $\chi$ is $\frac{1}{m}\sum_{k=1}^{m}\chi(g^k)$. Using this and a simple computation yields the claim.