Representations of $\operatorname{PSL}_2 (\Bbb{F}_p)$

Asked Jun 19 '24 at 23:21

Active Jun 20 '24 at 14:50

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Let $p>2$ be prime and consider the group $G=\operatorname{PSL}_2 (\Bbb{F}_p):=\operatorname{SL}_2 (\Bbb{F}_p)/\langle -I \rangle$. Find all complex irreducible representations of $G$.

I am aware of a general classification of irreducible representations of a semi-direct product $A\ltimes N$ where $A$ is Abelian. However, I could not find a fitting group action.

edited Jun 20 '24 at 14:50

asked Jun 19 '24 at 23:21

Robert

There are some nice notes by Mark Reeder that go over the more general $PGL(2,q)$ case. But for some reason I can't find a direct link to them. – Steve D Jun 19 '24 at 23:47
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$\operatorname{PGL}_2(\mathbb F_p)$ is not isomorphic to $\operatorname{PSL}_2(\mathbb F_p)$. Which one do you really want? – Just a user Jun 20 '24 at 07:52
@Justauser $\operatorname{PSL}_2 (\Bbb{F}_p)$. I edited it now. – Robert Jun 20 '24 at 14:50
Galois already knew $\operatorname{PSL}_2(\mathbb F_p)$ is simple for $p>3$, so to construct the representations through semidirect product won't work. I don't know if an answer of reasonable length can be written down here. It was an exercise in Fulton & Harris, after they worked out the representations of $\operatorname{GL}_2(\mathbb F_q)$ and/or $\operatorname{SL}_2(\mathbb F_q)$. – Just a user Jun 21 '24 at 07:15
@Justauser Thank you – Robert Jun 21 '24 at 13:02

Representations of $\operatorname{PSL}_2 (\Bbb{F}_p)$

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