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Let $\theta_1$ and $\theta_2$ be the fundamental weights of $\mathfrak{sl}(3,\mathbb{C})$. I know that for each pair $(a,b)\in\mathbb{Z}$ there is an unique irreducible representation $\Gamma_{a,b}$ (Fulton Harris notation) with maximal weight $a\theta_1+b\theta_2$. Furhtermore, I also know that $\Gamma_{a,b}$ is a subrepresentation of $\operatorname{Sym}^a\mathbb{C}^3\otimes\operatorname{Sym}^b(\mathbb{C}^3)^*$. It seems to me that if $\Gamma_{a,b}$ is self-conjugate, then $a=b$, is this correct?

But even if that is correct I still have some doubts. Are the representations $\Gamma_{a,a}$ of real type? And if $a\neq b$, can we ensure that $\Gamma_{a,b}$ is of complex type?

Cameron L. Williams
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Gomes93
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    It makes no sense to ask for representations of complex Lie algebras, like $\mathfrak{sl}3(\mathbb C)$, whether they are of real or complex type; you can ask that for reps of a real Lie algebra. Here it would depend on what real form of $\mathfrak{sl}_3(\mathbb C)$ you are originally interested in. In case it is the compact $\mathfrak{su}_3$, see here: https://math.stackexchange.com/q/2767862/96384, which in this case would confirm your suspicion: $-w_0$ flips $\theta_1 \leftrightarrow \theta_2$ and hence $a \leftrightarrow b$; thus, $\Gamma{a,b}$ is real or pseudoreal iff $a=b$. – Torsten Schoeneberg Jan 21 '21 at 20:46
  • @TorstenSchoeneberg can't a representation of complex Lie algebra have a real structure map? J:V\to V s.t. J^2=1? – Gomes93 Jan 22 '21 at 03:47
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    A real structure map has not only the property you write, but also, crucially, it has to be antilinear i.e. $J(\lambda v) = \bar{\lambda}v$ for all $\lambda \in \mathbb C$, and it has to be equivariant w.r.t. to the action of our Lie algebra $\mathfrak g$. Now try for yourself that if $\mathfrak g$ is a $\mathbb C$- (as opposed to a $\mathbb R$-) vector space, these two conditions give a contradiction unless the $\mathfrak g$-action is trivial. – Torsten Schoeneberg Jan 22 '21 at 05:43

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