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Annie
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Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$
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Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$
@YCor: That's true. I'm slightly less certain that there are no further examples when $m$ or $n$ is $1$. For instance, as stated there are probably no further examples if you replace $GL$ with $SL$, but if $m$ or $n$ is $1$ you can find new examples for $SL$.
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Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$
@Nate: Great observations! I sent you an email about this.
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Extending representations of $\mathrm{GL}(n,\mathbb{F}_p) \times \mathrm{GL}(m,\mathbb{F}_p)$ to $\mathrm{GL}(n+m,\mathbb{F}_p)$