I need to know if there are any 12 dimensional irreducible representations of $\mathfrak{sl}(3,\mathbb{C})$

I roughly understand the general theory of how every finite dimensional representation has a decomposition into weight spaces and so on (Fulton Harris). I am having difficulty in explicitly computing it for this case. Any help or direction (like a procedure to follow) to arrive at the result will be much appreciated. Thanks a lot!

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    $\begingroup$ Do you know what the representations "look like" in the weight spaces? For example, do you know what the fundamental rep and the adjoint rep look like? Also, are you aware that the dimension and weights of a representation are completely determined by the highest weight? And do you know how to compute these weights, given the highest weight? $\endgroup$ – Kenny Wong Oct 19 '17 at 14:12
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    $\begingroup$ There is even a formula for the dimension of an irrep, given in terms of the highest weights: math.stackexchange.com/questions/631654/…. $\endgroup$ – Kenny Wong Oct 19 '17 at 14:14
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    $\begingroup$ But if you know this, then it reduces to a number theory problem: Do there exist non-negative integers $a$, $b$ such that $\frac 1 2 (a + 1)(b+1)(a + b + 2) = 12$? $\endgroup$ – Kenny Wong Oct 19 '17 at 14:23
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    $\begingroup$ And my point is that $a + 1$, $b+1$ and $a+ b + 2$ are all positive factors of 24. So $a + 1$, $b + 1$ and $a + b + 2$ must all be contained in the set $\{1, 2, 3, 4, 6, 12, 24\}$. $\endgroup$ – Kenny Wong Oct 19 '17 at 15:07
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    $\begingroup$ Well, first of all, since $x + y \geq x$, we must have $x \leq \sqrt{24}$, i.e. $x \in \{ 1, 2, 3, 4 \}$. Same for $y$. So you don't exactly have many cases to check... $\endgroup$ – Kenny Wong Oct 19 '17 at 15:41

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