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This exercise asks you to build the character table of $\operatorname{SL}(2,\mathbb{Z}/3)$. I have found numerous resources that solve the problem in sophisticated ways, and for more general $F_p$ field.

However, given that this problem is posed in chapter 3 of Fulton & Harris, I was wondering whether there is a simple way to solve it with the methods covered so far.

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    Have you calculated the derived subgroup? – 2017-01-12
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    Sorry, I'm only a physicist. But again, the derived subgroup also sounds more sophisticated than what you'd expect in chapter 3 of Fulton & Harris. They haven't even introduced induced representations at that point in the book. Clearly there must be a simpler way. – 2017-01-13
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    I'd also like to see some ideas for how to do this. I figured out all the 1 dimensional and the 3 dimensional representation, and a little staring will convince you the 3 remaining representations are all dimension 2. But how to get them? – 2018-11-10

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