Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h}$ a CSA with root system $\Phi$, base $\Delta$, and Weyl group $W$. Then there exists a unique element $\sigma\in W$ such that $\sigma(\Delta)=-\Delta$.

Write given a dominant integral weight $\lambda$, write $V(\lambda)$ for the unique finite dimensional irreducible representation of $\mathfrak{g}$ of maximal weight $\lambda$. Then one can show that $V(\lambda)^*\cong V(-\sigma\lambda)$, where $V(\lambda)^*$ is the dual representation.

It follows that $V(\lambda)$ is self dual if and only if $\sigma\lambda=-\lambda$. For this reason I'm interested in computing $\sigma$ explicitly for various cases so that I can find its $(-1)$ eigenspace and intersect that with the dominant integral weights. For $\mathfrak{sl}_3$ with the 'standard' CSA and base, I believe that $\sigma=\begin{bmatrix}0 & -1\\-1 & 0\end{bmatrix}$ with respect to the basis $\{\lambda_1,\lambda_2\}$ of $\mathfrak{h}$ consisting of the fundamental dominant weights. But this was easy to compute because the Weyl group is small. In general $\sigma$ seems difficult to compute. For instance, if we were to write it as a product of simple reflections (i.e. the reflections from simple roots) we would need at least $|\Phi|/2$ many terms to write it. And even if we find a finite order orthogonal transformation $\sigma$ such that $\sigma(\Delta)=-\Delta$, it may not be in the Weyl group.

So my question is, are there clever methods of finding $\sigma$, or checking whether or not a transformation is in the Weyl group?

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    $\begingroup$ Your $\sigma$ is the so called «longest elemen» of th Weeyl group. If you google for that you'll find tons of information. $\endgroup$ – Mariano Suárez-Álvarez Jul 17 '16 at 5:56
  • $\begingroup$ Oh, great, thanks! $\endgroup$ – freeRmodule Jul 18 '16 at 1:47

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